Multiply the following polynomials: $$ \begin{array}{ll} ext { 15. }\left(8 x^{3} y^{2} ight)\left(-3 x^{2} y^{3} ight) & ext { 25. }(4 x-3)(3 x-5) \ ext { 16. }\left(-9 x^{3} y ight)\left(-8 x^{2} y^{3} ight) & ext { 26. }(x-8)(x-7) \ ext { 17. } j^{2}\left(k^{5} j^{3} ight) & ext { 27. }(6 a+1)(5 a+2) \ ext { 18. } a^{4}\left(b^{4} a^{6} ight) & ext { 28. }(5 x+4 y)(2 x+5 y) \ ext { 19. } 2 x^{3}\left(9 x^{2}+5 y ight) & ext { 29. }(2 x+y)(4 x-9 y) \ ext { 20. } 5 x^{3}(2 x+4 y) & ext { 30. }(6 r-5)(6 r+1) \ ext { 21. } 5 m^{2}\left(3 m^{3}+5 m^{2}-4 m+6 ight) & ext { 31. }(6 c+7)(6 c-7) \ ext { 22. }-4 x^{2} y\left(x^{2}+7 x y-6 y^{3} ight) & ext { 32. }(3 x+5 y)^{2} \ ext { 23. }(x+6)(x+2) & ext { 33. }(x-2)\left(x^{2}-x+3 ight) \ ext { 24. }(x-6)(x+9) & 34 \cdot(2 x-5)\left(5 x^{2}+4 x+7 ight)\end{array} $$

Question

Multiply the following polynomials: $$ \begin{array}{ll}	ext { 15. }\left(8 x^{3} y^{2}ight)\left(-3 x^{2} y^{3}ight) & 	ext { 25. }(4 x-3)(3 x-5) \ 	ext { 16. }\left(-9 x^{3} yight)\left(-8 x^{2} y^{3}ight) & 	ext { 26. }(x-8)(x-7) \ 	ext { 17. } j^{2}\left(k^{5} j^{3}ight) & 	ext { 27. }(6 a+1)(5 a+2) \ 	ext { 18. } a^{4}\left(b^{4} a^{6}ight) & 	ext { 28. }(5 x+4 y)(2 x+5 y) \ 	ext { 19. } 2 x^{3}\left(9 x^{2}+5 yight) & 	ext { 29. }(2 x+y)(4 x-9 y) \ 	ext { 20. } 5 x^{3}(2 x+4 y) & 	ext { 30. }(6 r-5)(6 r+1) \ 	ext { 21. } 5 m^{2}\left(3 m^{3}+5 m^{2}-4 m+6ight) & 	ext { 31. }(6 c+7)(6 c-7) \ 	ext { 22. }-4 x^{2} y\left(x^{2}+7 x y-6 y^{3}ight) & 	ext { 32. }(3 x+5 y)^{2} \ 	ext { 23. }(x+6)(x+2) & 	ext { 33. }(x-2)\left(x^{2}-x+3ight) \ 	ext { 24. }(x-6)(x+9) & 34 \cdot(2 x-5)\left(5 x^{2}+4 x+7ight)\end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Simplify:
  $$5x^{3}\left(2x+4yight)$$
- step1: Rewrite the expression:
  $$5\left(2x+4yight)x^{3}$$
- step2: Multiply the expression:
  $$10\left(x+2yight)x^{3}$$
- step3: Rearrange the terms:
  $$10x^{4}+20yx^{3}$$
Expand the expression $$ (2 x-5)(5 x^{2}+4 x+7) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(2x-5ight)\left(5x^{2}+4x+7ight)$$
- step1: Apply the distributive property:
  $$2x	imes 5x^{2}+2x	imes 4x+2x	imes 7-5	imes 5x^{2}-5	imes 4x-5	imes 7$$
- step2: Multiply the terms:
  $$10x^{3}+8x^{2}+14x-25x^{2}-20x-35$$
- step3: Subtract the terms:
  $$10x^{3}-17x^{2}-6x-35$$
Expand the expression $$ j^{2}(k^{5} j^{3}) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$j^{2}\left(k^{5}j^{3}ight)$$
- step1: Multiply the terms:
  $$j^{2}k^{5}j^{3}$$
- step2: Multiply the terms:
  $$j^{5}k^{5}$$
Expand the expression $$ (8 x^{3} y^{2})(-3 x^{2} y^{3}) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$\left(8x^{3}y^{2}ight)\left(-3x^{2}y^{3}ight)$$
- step1: Calculate:
  $$8x^{3}y^{2}\left(-3x^{2}y^{3}ight)$$
- step2: Multiply the numbers:
  $$-24x^{3}y^{2}x^{2}y^{3}$$
- step3: Multiply the terms:
  $$-24x^{5}y^{2}	imes y^{3}$$
- step4: Multiply the terms:
  $$-24x^{5}y^{5}$$
Expand the expression $$ 5 m^{2}(3 m^{3}+5 m^{2}-4 m+6) $$
Simplify the expression by following steps:
- step0: Calculate:
  $$5m^{2}\left(3m^{3}+5m^{2}-4m+6ight)$$
- step1: Rewrite the expression:
  $$5\left(3m^{3}+5m^{2}-4m+6ight)m^{2}$$
- step2: Rearrange the terms:
  $$\left(3m^{3}+5m^{2}-4m+6ight)	imes 5m^{2}$$
- step3: Apply the distributive property:
  $$3m^{3}	imes 5m^{2}+5m^{2}	imes 5m^{2}-4m	imes 5m^{2}+6	imes 5m^{2}$$
- step4: Multiply the terms:
  $$15m^{5}+25m^{4}-20m^{3}+30m^{2}$$
Expand the expression $$ (6 a+1)(5 a+2) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(6a+1ight)\left(5a+2ight)$$
- step1: Apply the distributive property:
  $$6a	imes 5a+6a	imes 2+1	imes 5a+1	imes 2$$
- step2: Multiply the terms:
  $$30a^{2}+12a+5a+2$$
- step3: Add the terms:
  $$30a^{2}+17a+2$$
Expand the expression $$ (x+6)(x+2) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x+6ight)\left(x+2ight)$$
- step1: Apply the distributive property:
  $$x	imes x+x	imes 2+6x+6	imes 2$$
- step2: Multiply the terms:
  $$x^{2}+2x+6x+12$$
- step3: Add the terms:
  $$x^{2}+8x+12$$
Expand the expression $$ -4 x^{2} y(x^{2}+7 x y-6 y^{3}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$-4x^{2}y\left(x^{2}+7xy-6y^{3}ight)$$
- step1: Apply the distributive property:
  $$-4x^{2}yx^{2}-4x^{2}y	imes 7xy-\left(-4x^{2}y	imes 6y^{3}ight)$$
- step2: Multiply the terms:
  $$-4x^{4}y-28x^{3}y^{2}-\left(-24x^{2}y^{4}ight)$$
- step3: Remove the parentheses:
  $$-4x^{4}y-28x^{3}y^{2}+24x^{2}y^{4}$$
Expand the expression $$ (x-6)(x+9) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x-6ight)\left(x+9ight)$$
- step1: Apply the distributive property:
  $$x	imes x+x	imes 9-6x-6	imes 9$$
- step2: Multiply the terms:
  $$x^{2}+9x-6x-54$$
- step3: Subtract the terms:
  $$x^{2}+3x-54$$
Expand the expression $$ (4 x-3)(3 x-5) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(4x-3ight)\left(3x-5ight)$$
- step1: Apply the distributive property:
  $$4x	imes 3x-4x	imes 5-3	imes 3x-\left(-3	imes 5ight)$$
- step2: Multiply the terms:
  $$12x^{2}-20x-9x-\left(-15ight)$$
- step3: Remove the parentheses:
  $$12x^{2}-20x-9x+15$$
- step4: Subtract the terms:
  $$12x^{2}-29x+15$$
Expand the expression $$ (x-8)(x-7) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x-8ight)\left(x-7ight)$$
- step1: Apply the distributive property:
  $$x	imes x-x	imes 7-8x-\left(-8	imes 7ight)$$
- step2: Multiply the terms:
  $$x^{2}-7x-8x-\left(-56ight)$$
- step3: Remove the parentheses:
  $$x^{2}-7x-8x+56$$
- step4: Subtract the terms:
  $$x^{2}-15x+56$$
Expand the expression $$ a^{4}(b^{4} a^{6}) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$a^{4}\left(b^{4}a^{6}ight)$$
- step1: Multiply the terms:
  $$a^{4}b^{4}a^{6}$$
- step2: Multiply the terms:
  $$a^{10}b^{4}$$
Expand the expression $$ 2 x^{3}(9 x^{2}+5 y) $$
Simplify the expression by following steps:
- step0: Calculate:
  $$2x^{3}\left(9x^{2}+5yight)$$
- step1: Rewrite the expression:
  $$2\left(9x^{2}+5yight)x^{3}$$
- step2: Rearrange the terms:
  $$\left(9x^{2}+5yight)	imes 2x^{3}$$
- step3: Apply the distributive property:
  $$9x^{2}	imes 2x^{3}+5y	imes 2x^{3}$$
- step4: Multiply the terms:
  $$18x^{5}+10yx^{3}$$
Expand the expression $$ (6 r-5)(6 r+1) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(6r-5ight)\left(6r+1ight)$$
- step1: Apply the distributive property:
  $$6r	imes 6r+6r	imes 1-5	imes 6r-5	imes 1$$
- step2: Multiply the terms:
  $$36r^{2}+6r-30r-5$$
- step3: Subtract the terms:
  $$36r^{2}-24r-5$$
Expand the expression $$ (5 x+4 y)(2 x+5 y) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(5x+4yight)\left(2x+5yight)$$
- step1: Apply the distributive property:
  $$5x	imes 2x+5x	imes 5y+4y	imes 2x+4y	imes 5y$$
- step2: Multiply the terms:
  $$10x^{2}+25xy+8yx+20y^{2}$$
- step3: Add the terms:
  $$10x^{2}+33xy+20y^{2}$$
Expand the expression $$ (-9 x^{3} y)(-8 x^{2} y^{3}) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$\left(-9x^{3}yight)\left(-8x^{2}y^{3}ight)$$
- step1: Remove the parentheses:
  $$-9x^{3}y\left(-8x^{2}y^{3}ight)$$
- step2: Multiply the numbers:
  $$72x^{3}yx^{2}y^{3}$$
- step3: Multiply the terms:
  $$72x^{5}y	imes y^{3}$$
- step4: Multiply the terms:
  $$72x^{5}y^{4}$$
Expand the expression $$ (2 x+y)(4 x-9 y) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(2x+yight)\left(4x-9yight)$$
- step1: Apply the distributive property:
  $$2x	imes 4x-2x	imes 9y+y	imes 4x-y	imes 9y$$
- step2: Multiply the terms:
  $$8x^{2}-18xy+4yx-9y^{2}$$
- step3: Add the terms:
  $$8x^{2}-14xy-9y^{2}$$
Expand the expression $$ (3 x+5 y)^{2 $$
Simplify the expression by following steps:
- step0: Calculate:
  $$\left(3x+5yight)^{2}$$
- step1: Calculate:
  $$9x^{2}+30xy+25y^{2}$$
Expand the expression $$ (6 c+7)(6 c-7) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(6c+7ight)\left(6c-7ight)$$
- step1: Simplify the product:
  $$\left(6cight)^{2}-7^{2}$$
- step2: Evaluate the power:
  $$36c^{2}-49$$
Expand the expression $$ (x-2)(x^{2}-x+3) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x-2ight)\left(x^{2}-x+3ight)$$
- step1: Apply the distributive property:
  $$x	imes x^{2}-x	imes x+x	imes 3-2x^{2}-\left(-2xight)-2	imes 3$$
- step2: Multiply the terms:
  $$x^{3}-x^{2}+3x-2x^{2}-\left(-2xight)-6$$
- step3: Remove the parentheses:
  $$x^{3}-x^{2}+3x-2x^{2}+2x-6$$
- step4: Subtract the terms:
  $$x^{3}-3x^{2}+5x-6$$
Here are the results of the polynomial multiplications:

15. $$ (8 x^{3} y^{2})(-3 x^{2} y^{3}) = -24x^{5}y^{5} $$

16. $$ (-9 x^{3} y)(-8 x^{2} y^{3}) = 72x^{5}y^{4} $$

17. $$ j^{2}(k^{5} j^{3}) = j^{5}k^{5} $$

18. $$ a^{4}(b^{4} a^{6}) = a^{10}b^{4} $$

19. $$ 2 x^{3}(9 x^{2}+5 y) = 18x^{5}+10yx^{3} $$

20. $$ 5 x^{3}(2 x+4 y) = 10x^{4}+20yx^{3} $$

21. $$ 5 m^{2}(3 m^{3}+5 m^{2}-4 m+6) = 15m^{5}+25m^{4}-20m^{3}+30m^{2} $$

22. $$ -4 x^{2} y(x^{2}+7 x y-6 y^{3}) = -4x^{4}y-28x^{3}y^{2}+24x^{2}y^{4} $$

23. $$ (x+6)(x+2) = x^{2}+8x+12 $$

24. $$ (x-6)(x+9) = x^{2}+3x-54 $$

25. $$ (4 x-3)(3 x-5) = 12x^{2}-29x+15 $$

26. $$ (x-8)(x-7) = x^{2}-15x+56 $$

27. $$ (6 a+1)(5 a+2) = 30a^{2}+17a+2 $$

28. $$ (5 x+4 y)(2 x+5 y) = 10x^{2}+33xy+20y^{2} $$

29. $$ (2 x+y)(4 x-9 y) = 8x^{2}-14xy-9y^{2} $$

30. $$ (6 r-5)(6 r+1) = 36r^{2}-24r-5 $$

31. $$ (6 c+7)(6 c-7) = 36c^{2}-49 $$

32. $$ (3 x+5 y)^{2} = 9x^{2}+30xy+25y^{2} $$

33. $$ (x-2)(x^{2}-x+3) = x^{3}-3x^{2}+5x-6 $$

34. $$ (2 x-5)(5 x^{2}+4 x+7) = 10x^{3}-17x^{2}-6x-35 $$

If you need further assistance or explanations for any specific multiplication, feel free to ask!
Here are the results of the polynomial multiplications: 15. $$ -24x^{5}y^{5} $$ 16. $$ 72x^{5}y^{4} $$ 17. $$ j^{5}k^{5} $$ 18. $$ a^{10}b^{4} $$ 19. $$ 18x^{5}+10yx^{3} $$ 20. $$ 10x^{4}+20yx^{3} $$ 21. $$ 15m^{5}+25m^{4}-20m^{3}+30m^{2} $$ 22. $$ -4x^{4}y-28x^{3}y^{2}+24x^{2}y^{4} $$ 23. $$ x^{2}+8x+12 $$ 24. $$ x^{2}+3x-54 $$ 25. $$ 12x^{2}-29x+15 $$ 26. $$ x^{2}-15x+56 $$ 27. $$ 30a^{2}+17a+2 $$ 28. $$ 10x^{2}+33xy+20y^{2} $$ 29. $$ 8x^{2}-14xy-9y^{2} $$ 30. $$ 36r^{2}-24r-5 $$ 31. $$ 36c^{2}-49 $$ 32. $$ 9x^{2}+30xy+25y^{2} $$ 33. $$ x^{3}-3x^{2}+5x-6 $$ 34. $$ 10x^{3}-17x^{2}-6x-35 $$ If you need further assistance or explanations for any specific multiplication, feel free to ask!

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