Question 2 2. Simplify: 2.1. \( (2 y-3)\left(6 y^{2}+4 y-8\right) \) 2.2. \( \left(\frac{1}{p}-q\right)\left(\frac{1}{p}+q\right)-\frac{q}{p^{2}}\left(\frac{1}{p}+q p^{2}\right) \) 2.3. \( \frac{x^{3}-27}{x^{2}+3 x+9} \) 2.4. \( \frac{(3 x)^{2}(-2 x y)^{3}}{2 x^{5} y^{4}} \) 2.5. \( \left(\frac{a}{2}+1\right)\left(\frac{a}{2}-1\right) \) 2.6. \( (a-2)\left(a^{2}+2 a+4\right) \) 2.7. \( \frac{3 x^{3}-7 x^{2}-6 x}{x^{2}-9} \div \frac{3 x^{2}+2 x}{3 x+9} \)

Simplify the expression by following steps: - step0: Multiply the terms: \(\left(2y-3\right)\left(6y^{2}+4y-8\right)\) - step1: Apply the distributive property: \(2y\times 6y^{2}+2y\times 4y-2y\times 8-3\times 6y^{2}-3\times 4y-\left(-3\times 8\right)\) - step2: Multiply the terms: \(12y^{3}+8y^{2}-16y-18y^{2}-12y-\left(-24\right)\) - step3: Remove the parentheses: \(12y^{3}+8y^{2}-16y-18y^{2}-12y+24\) - step4: Subtract the terms: \(12y^{3}-10y^{2}-28y+24\) Expand the expression \( \frac{3 x^{3}-7 x^{2}-6 x}{x^{2}-9} \div \frac{3 x^{2}+2 x}{3 x+9} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{3x^{3}-7x^{2}-6x}{x^{2}-9}\div \frac{3x^{2}+2x}{3x+9}\) - step1: Divide the terms: \(\frac{3x^{2}+2x}{x+3}\div \frac{3x^{2}+2x}{3x+9}\) - step2: Multiply by the reciprocal: \(\frac{3x^{2}+2x}{x+3}\times \frac{3x+9}{3x^{2}+2x}\) - step3: Reduce the fraction: \(\frac{1}{x+3}\times \left(3x+9\right)\) - step4: Rewrite the expression: \(\frac{1}{x+3}\times 3\left(x+3\right)\) - step5: Reduce the fraction: \(1\times 3\) - step6: Multiply the terms: \(3\) Expand the expression \( \frac{x^{3}-27}{x^{2}+3 x+9} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{x^{3}-27}{x^{2}+3x+9}\) - step1: Calculate: \(\frac{\left(x-3\right)\left(x^{2}+3x+9\right)}{x^{2}+3x+9}\) - step2: Reduce the fraction: \(x-3\) Expand the expression \( \left(\frac{a}{2}+1\right)\left(\frac{a}{2}-1\right) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(\frac{a}{2}+1\right)\left(\frac{a}{2}-1\right)\) - step1: Add the terms: \(\frac{a+2}{2}\times \left(\frac{a}{2}-1\right)\) - step2: Subtract the terms: \(\frac{a+2}{2}\times \frac{a-2}{2}\) - step3: Multiply the terms: \(\frac{\left(a+2\right)\left(a-2\right)}{2\times 2}\) - step4: Multiply the terms: \(\frac{\left(a+2\right)\left(a-2\right)}{4}\) - step5: Multiply the terms: \(\frac{a^{2}-4}{4}\) Expand the expression \( \frac{(3 x)^{2}(-2 x y)^{3}}{2 x^{5} y^{4}} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{\left(3x\right)^{2}\left(-2xy\right)^{3}}{2x^{5}y^{4}}\) - step1: Multiply the terms: \(\frac{-72x^{5}y^{3}}{2x^{5}y^{4}}\) - step2: Reduce the fraction: \(\frac{-72y^{3}}{2y^{4}}\) - step3: Reduce the fraction: \(\frac{-72}{2y^{4-3}}\) - step4: Reduce the fraction: \(\frac{-72}{2y}\) - step5: Divide the terms: \(\frac{-36}{y}\) - step6: Rewrite the fraction: \(-\frac{36}{y}\) Expand the expression \( (a-2)(a^{2}+2 a+4) \) Simplify the expression by following steps: - step0: Simplify the product: \(\left(a-2\right)\left(a^{2}+2a+4\right)\) - step1: Simplify the product: \(a^{3}-8\) Expand the expression \( \left(\frac{1}{p}-q\right)\left(\frac{1}{p}+q\right)-\frac{q}{p^{2}}\left(\frac{1}{p}+q p^{2}\right) \) Simplify the expression by following steps: - step0: Simplify: \(\left(\frac{1}{p}-q\right)\left(\frac{1}{p}+q\right)-\frac{q}{p^{2}}\times \left(\frac{1}{p}+qp^{2}\right)\) - step1: Subtract the terms: \(\frac{1-qp}{p}\times \left(\frac{1}{p}+q\right)-\frac{q}{p^{2}}\times \left(\frac{1}{p}+qp^{2}\right)\) - step2: Add the terms: \(\frac{1-qp}{p}\times \frac{1+qp}{p}-\frac{q}{p^{2}}\times \left(\frac{1}{p}+qp^{2}\right)\) - step3: Add the terms: \(\frac{1-qp}{p}\times \frac{1+qp}{p}-\frac{q}{p^{2}}\times \frac{1+qp^{3}}{p}\) - step4: Multiply the terms: \(\frac{\left(1-qp\right)\left(1+qp\right)}{p^{2}}-\frac{q}{p^{2}}\times \frac{1+qp^{3}}{p}\) - step5: Multiply the terms: \(\frac{\left(1-qp\right)\left(1+qp\right)}{p^{2}}-\frac{q\left(1+qp^{3}\right)}{p^{3}}\) - step6: Reduce fractions to a common denominator: \(\frac{\left(1-qp\right)\left(1+qp\right)p}{p^{2}\times p}-\frac{q\left(1+qp^{3}\right)}{p^{3}}\) - step7: Multiply: \(\frac{\left(1-qp\right)\left(1+qp\right)p}{p^{3}}-\frac{q\left(1+qp^{3}\right)}{p^{3}}\) - step8: Transform the expression: \(\frac{\left(1-qp\right)\left(1+qp\right)p-q\left(1+qp^{3}\right)}{p^{3}}\) - step9: Multiply the terms: \(\frac{p-q^{2}p^{3}-q\left(1+qp^{3}\right)}{p^{3}}\) - step10: Multiply the terms: \(\frac{p-q^{2}p^{3}-\left(q+q^{2}p^{3}\right)}{p^{3}}\) - step11: Subtract the terms: \(\frac{p-2q^{2}p^{3}-q}{p^{3}}\) Here are the simplified results for each part of the question: ### 2.1 Simplifying \( (2 y-3)(6 y^{2}+4 y-8) \): \[ 12y^{3}-10y^{2}-28y+24 \] ### 2.2 Simplifying \( \left(\frac{1}{p}-q\right)\left(\frac{1}{p}+q\right)-\frac{q}{p^{2}}\left(\frac{1}{p}+q p^{2}\right) \): \[ \frac{p-2q^{2}p^{3}-q}{p^{3}} \] ### 2.3 Simplifying \( \frac{x^{3}-27}{x^{2}+3 x+9} \): \[ x-3 \] ### 2.4 Simplifying \( \frac{(3 x)^{2}(-2 x y)^{3}}{2 x^{5} y^{4}} \): \[ -\frac{36}{y} \] ### 2.5 Simplifying \( \left(\frac{a}{2}+1\right)\left(\frac{a}{2}-1\right) \): \[ \frac{a^{2}-4}{4} \] ### 2.6 Simplifying \( (a-2)(a^{2}+2 a+4) \): \[ a^{3}-8 \] ### 2.7 Simplifying \( \frac{3 x^{3}-7 x^{2}-6 x}{x^{2}-9} \div \frac{3 x^{2}+2 x}{3 x+9} \): \[ 3 \] If you have any further questions or need additional assistance, feel free to ask!