$$ \begin{array}{ll}(x+3)\left(x^{2}+x+2 ight) & ext { 2. }(a+b)\left(a^{2}-a b+b^{2} ight) \ (x+2)\left(3 x^{2}-7 x-1 ight) & ext { 5. }(3 x+5)\left(2 x^{2}-x-2 ight) \ \left(x+\frac{1}{x} ight)\left(x^{2}-1+\frac{1}{x^{2}} ight) & ext { 8. }(a-b)(a+b)^{2}\end{array} $$

Question

$$ \begin{array}{ll}(x+3)\left(x^{2}+x+2ight) & 	ext { 2. }(a+b)\left(a^{2}-a b+b^{2}ight) \ (x+2)\left(3 x^{2}-7 x-1ight) & 	ext { 5. }(3 x+5)\left(2 x^{2}-x-2ight) \ \left(x+\frac{1}{x}ight)\left(x^{2}-1+\frac{1}{x^{2}}ight) & 	ext { 8. }(a-b)(a+b)^{2}\end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Simplify the product:
  $$\left(a+bight)\left(a^{2}-ab+b^{2}ight)$$
- step1: Simplify the product:
  $$a^{3}+b^{3}$$
Expand the expression $$ (x+3)(x^{2}+x+2) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x+3ight)\left(x^{2}+x+2ight)$$
- step1: Apply the distributive property:
  $$x	imes x^{2}+x	imes x+x	imes 2+3x^{2}+3x+3	imes 2$$
- step2: Multiply the terms:
  $$x^{3}+x^{2}+2x+3x^{2}+3x+6$$
- step3: Add the terms:
  $$x^{3}+4x^{2}+5x+6$$
Expand the expression $$ (x+2)(3x^{2}-7x-1) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x+2ight)\left(3x^{2}-7x-1ight)$$
- step1: Apply the distributive property:
  $$x	imes 3x^{2}-x	imes 7x-x	imes 1+2	imes 3x^{2}-2	imes 7x-2	imes 1$$
- step2: Multiply the terms:
  $$3x^{3}-7x^{2}-x+6x^{2}-14x-2$$
- step3: Add the terms:
  $$3x^{3}-x^{2}-15x-2$$
Expand the expression $$ (3x+5)(2x^{2}-x-2) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(3x+5ight)\left(2x^{2}-x-2ight)$$
- step1: Apply the distributive property:
  $$3x	imes 2x^{2}-3x	imes x-3x	imes 2+5	imes 2x^{2}-5x-5	imes 2$$
- step2: Multiply the terms:
  $$6x^{3}-3x^{2}-6x+10x^{2}-5x-10$$
- step3: Add the terms:
  $$6x^{3}+7x^{2}-11x-10$$
Expand the expression $$ (a-b)(a+b)^{2} $$
Simplify the expression by following steps:
- step0: Expand the expression:
  $$\left(a-bight)\left(a+bight)^{2}$$
- step1: Calculate:
  $$\left(a-bight)\left(a^{2}+2ab+b^{2}ight)$$
- step2: Apply the distributive property:
  $$a	imes a^{2}+a	imes 2ab+ab^{2}-ba^{2}-b	imes 2ab-b	imes b^{2}$$
- step3: Multiply the terms:
  $$a^{3}+2a^{2}b+ab^{2}-ba^{2}-2b^{2}a-b^{3}$$
- step4: Subtract the terms:
  $$a^{3}+a^{2}b-ab^{2}-b^{3}$$
Expand the expression $$ (x+\frac{1}{x})(x^{2}-1+\frac{1}{x^{2}}) $$
Simplify the expression by following steps:
- step0: Simplify the product:
  $$\left(x+\frac{1}{x}ight)\left(x^{2}-1+\frac{1}{x^{2}}ight)$$
- step1: Add the terms:
  $$\frac{x^{2}+1}{x}	imes \left(x^{2}-1+\frac{1}{x^{2}}ight)$$
- step2: Calculate:
  $$\frac{x^{2}+1}{x}	imes \frac{x^{4}-x^{2}+1}{x^{2}}$$
- step3: Multiply the terms:
  $$\frac{\left(x^{2}+1ight)\left(x^{4}-x^{2}+1ight)}{x	imes x^{2}}$$
- step4: Multiply the terms:
  $$\frac{\left(x^{2}+1ight)\left(x^{4}-x^{2}+1ight)}{x^{3}}$$
- step5: Simplify the product:
  $$\frac{x^{6}+1}{x^{3}}$$
Here are the expanded forms for each of the expressions provided:

1. **Expression:** $$ (x+3)(x^{2}+x+2) $$  
   **Expanded Form:** $$ x^{3}+4x^{2}+5x+6 $$

2. **Expression:** $$ (a+b)(a^{2}-ab+b^{2}) $$  
   **Expanded Form:** $$ a^{3}+b^{3} $$

3. **Expression:** $$ (x+2)(3x^{2}-7x-1) $$  
   **Expanded Form:** $$ 3x^{3}-x^{2}-15x-2 $$

4. **Expression:** $$ (3x+5)(2x^{2}-x-2) $$  
   **Expanded Form:** $$ 6x^{3}+7x^{2}-11x-10 $$

5. **Expression:** $$ (x+\frac{1}{x})(x^{2}-1+\frac{1}{x^{2}}) $$  
   **Expanded Form:** $$ \frac{x^{6}+1}{x^{3}} $$

6. **Expression:** $$ (a-b)(a+b)^{2} $$  
   **Expanded Form:** $$ a^{3}+a^{2}b-ab^{2}-b^{3} $$

If you have any further questions or need additional assistance, feel free to ask!
Here are the expanded forms of the given expressions: 1. $$ (x+3)(x^{2}+x+2) = x^{3}+4x^{2}+5x+6 $$ 2. $$ (a+b)(a^{2}-ab+b^{2}) = a^{3}+b^{3} $$ 3. $$ (x+2)(3x^{2}-7x-1) = 3x^{3}-x^{2}-15x-2 $$ 4. $$ (3x+5)(2x^{2}-x-2) = 6x^{3}+7x^{2}-11x-10 $$ 5. $$ \left(x+\frac{1}{x}ight)\left(x^{2}-1+\frac{1}{x^{2}}ight) = \frac{x^{6}+1}{x^{3}} $$ 6. $$ (a-b)(a+b)^{2} = a^{3}+a^{2}b-ab^{2}-b^{3} $$

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