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

Simplify the expression by following steps: - step0: Simplify the product: \(\left(a+b\right)\left(a^{2}-ab+b^{2}\right)\) - 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: \(\left(x+3\right)\left(x^{2}+x+2\right)\) - step1: Apply the distributive property: \(x\times x^{2}+x\times x+x\times 2+3x^{2}+3x+3\times 2\) - 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: \(\left(x+2\right)\left(3x^{2}-7x-1\right)\) - step1: Apply the distributive property: \(x\times 3x^{2}-x\times 7x-x\times 1+2\times 3x^{2}-2\times 7x-2\times 1\) - 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: \(\left(3x+5\right)\left(2x^{2}-x-2\right)\) - step1: Apply the distributive property: \(3x\times 2x^{2}-3x\times x-3x\times 2+5\times 2x^{2}-5x-5\times 2\) - 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: \(\left(a-b\right)\left(a+b\right)^{2}\) - step1: Calculate: \(\left(a-b\right)\left(a^{2}+2ab+b^{2}\right)\) - step2: Apply the distributive property: \(a\times a^{2}+a\times 2ab+ab^{2}-ba^{2}-b\times 2ab-b\times b^{2}\) - 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: \(\left(x+\frac{1}{x}\right)\left(x^{2}-1+\frac{1}{x^{2}}\right)\) - step1: Add the terms: \(\frac{x^{2}+1}{x}\times \left(x^{2}-1+\frac{1}{x^{2}}\right)\) - step2: Calculate: \(\frac{x^{2}+1}{x}\times \frac{x^{4}-x^{2}+1}{x^{2}}\) - step3: Multiply the terms: \(\frac{\left(x^{2}+1\right)\left(x^{4}-x^{2}+1\right)}{x\times x^{2}}\) - step4: Multiply the terms: \(\frac{\left(x^{2}+1\right)\left(x^{4}-x^{2}+1\right)}{x^{3}}\) - 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!