Expand and simplify: $$ \begin{array}{lll} ext { 1) } \ ext { 3) } \ ext { 3) } & (2 x+4)\left(x^{2}+2 x+3 ight) & ext { (2) } \ (x-1)\left(x^{2}-2 x+3 ight) \ ext { 5) } & (3 x-y)\left(2 x^{2}+4 x y-y^{2} ight) & ext { (4) } \ & ext { (6) }(2 x-4)\left(x^{2}-3 x+1 ight) \ & & \end{array} $$

Question

Expand and simplify: $$ \begin{array}{lll}	ext { 1) } \ 	ext { 3) } \ 	ext { 3) } & (2 x+4)\left(x^{2}+2 x+3ight) & 	ext { (2) } \ (x-1)\left(x^{2}-2 x+3ight) \ 	ext { 5) } & (3 x-y)\left(2 x^{2}+4 x y-y^{2}ight) & 	ext { (4) } \ & 	ext { (6) }(2 x-4)\left(x^{2}-3 x+1ight) \ & & \end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x-1\right)\left(x^{2}-2x+3\right)$$
- step1: Apply the distributive property:
  $$x\times x^{2}-x\times 2x+x\times 3-x^{2}-\left(-2x\right)-3$$
- step2: Multiply the terms:
  $$x^{3}-2x^{2}+3x-x^{2}-\left(-2x\right)-3$$
- step3: Remove the parentheses:
  $$x^{3}-2x^{2}+3x-x^{2}+2x-3$$
- step4: Subtract the terms:
  $$x^{3}-3x^{2}+5x-3$$
Expand the expression $$ (2 x+4)(x^{2}+2 x+3) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(2x+4\right)\left(x^{2}+2x+3\right)$$
- step1: Apply the distributive property:
  $$2x\times x^{2}+2x\times 2x+2x\times 3+4x^{2}+4\times 2x+4\times 3$$
- step2: Multiply the terms:
  $$2x^{3}+4x^{2}+6x+4x^{2}+8x+12$$
- step3: Add the terms:
  $$2x^{3}+8x^{2}+14x+12$$
Expand the expression $$ (2 x-4)(x^{2}-3 x+1) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(2x-4\right)\left(x^{2}-3x+1\right)$$
- step1: Apply the distributive property:
  $$2x\times x^{2}-2x\times 3x+2x\times 1-4x^{2}-\left(-4\times 3x\right)-4\times 1$$
- step2: Multiply the terms:
  $$2x^{3}-6x^{2}+2x-4x^{2}-\left(-12x\right)-4$$
- step3: Remove the parentheses:
  $$2x^{3}-6x^{2}+2x-4x^{2}+12x-4$$
- step4: Subtract the terms:
  $$2x^{3}-10x^{2}+14x-4$$
Expand the expression $$ (3 x-y)(2 x^{2}+4 x y-y^{2}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(3x-y\right)\left(2x^{2}+4xy-y^{2}\right)$$
- step1: Apply the distributive property:
  $$3x\times 2x^{2}+3x\times 4xy-3xy^{2}-y\times 2x^{2}-y\times 4xy-\left(-y\times y^{2}\right)$$
- step2: Multiply the terms:
  $$6x^{3}+12x^{2}y-3xy^{2}-2yx^{2}-4y^{2}x-\left(-y^{3}\right)$$
- step3: Remove the parentheses:
  $$6x^{3}+12x^{2}y-3xy^{2}-2yx^{2}-4y^{2}x+y^{3}$$
- step4: Subtract the terms:
  $$6x^{3}+10x^{2}y-7xy^{2}+y^{3}$$
Here are the expanded and simplified results for each expression:

1) For $$ (2x + 4)(x^2 + 2x + 3) $$:
   $$
   2x^{3} + 8x^{2} + 14x + 12
   $$

2) For $$ (x - 1)(x^2 - 2x + 3) $$:
   $$
   x^{3} - 3x^{2} + 5x - 3
   $$

3) For $$ (3x - y)(2x^2 + 4xy - y^2) $$:
   $$
   6x^{3} + 10x^{2}y - 7xy^{2} + y^{3}
   $$

4) For $$ (2x - 4)(x^2 - 3x + 1) $$:
   $$
   2x^{3} - 10x^{2} + 14x - 4
   $$

These results provide the expanded forms of the given expressions.
Here are the expanded and simplified results: 1) $$ (2x + 4)(x^2 + 2x + 3) = 2x^3 + 8x^2 + 14x + 12 $$ 2) $$ (x - 1)(x^2 - 2x + 3) = x^3 - 3x^2 + 5x - 3 $$ 3) $$ (3x - y)(2x^2 + 4xy - y^2) = 6x^3 + 10x^2y - 7xy^2 + y^3 $$ 4) $$ (2x - 4)(x^2 - 3x + 1) = 2x^3 - 10x^2 + 14x - 4 $$

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