1. **Problem:** Simplify the product $$\left(\frac{5y^3}{32x}\right) \times \left(\frac{-4}{15x^2 y^3}\right)$$ 2. **Step 1: Factor all numerators and denominators completely.** - Numerators: $5y^3$ and $-4$ - Denominators: $32x$ and $15x^2 y^3$ 3. **Step 2: Multiply numerators and denominators:** $$\frac{5y^3 \times (-4)}{32x \times 15x^2 y^3} = \frac{-20 y^3}{480 x^3 y^3}$$ 4. **Step 3: Divide out common factors:** - $y^3$ cancels out numerator and denominator. - $20$ divides into $480$ exactly 24 times. So, $$\frac{-20}{480 x^3} = \frac{-1}{24 x^3}$$ **Final answer:** $$\boxed{\frac{-1}{24 x^3}}$$ --- 2. **Problem:** Simplify $$\left(\frac{y^3}{8}\right) \times \left(\frac{9x^2}{y^3}\right)$$ 3. Multiply numerators and denominators: $$\frac{y^3 \times 9x^2}{8 \times y^3} = \frac{9x^2 y^3}{8 y^3}$$ 4. Cancel $y^3$: $$\frac{9x^2}{8}$$ **Final answer:** $$\boxed{\frac{9x^2}{8}}$$ --- 3. **Problem:** Simplify $$\left(\frac{1}{4x - 3}\right) \times (20x - 15)$$ 4. Factor $20x - 15$: $$5(4x - 3)$$ 5. Multiply: $$\frac{1}{4x - 3} \times 5(4x - 3) = 5$$ **Final answer:** $$\boxed{5}$$ --- 4. **Problem:** Simplify $$\left(\frac{x - 6}{2x + 5}\right) \times \left(\frac{2x}{-x + 6}\right)$$ 5. Note $-x + 6 = -(x - 6)$ 6. Multiply: $$\frac{x - 6}{2x + 5} \times \frac{2x}{-(x - 6)} = \frac{(x - 6) 2x}{(2x + 5)(-(x - 6))} = \frac{2x (x - 6)}{-(2x + 5)(x - 6)}$$ 7. Cancel $(x - 6)$: $$\frac{2x}{-(2x + 5)} = \frac{-2x}{2x + 5}$$ **Final answer:** $$\boxed{\frac{-2x}{2x + 5}}$$ --- 5. **Problem:** Simplify $$\left(\frac{x^3 + x}{5}\right) \times \left(\frac{10}{x^2 + x}\right)$$ 6. Factor numerators and denominators: - $x^3 + x = x(x^2 + 1)$ - $x^2 + x = x(x + 1)$ 7. Multiply: $$\frac{x(x^2 + 1)}{5} \times \frac{10}{x(x + 1)} = \frac{10 x (x^2 + 1)}{5 x (x + 1)}$$ 8. Cancel $x$ and simplify $\frac{10}{5} = 2$: $$\frac{2 (x^2 + 1)}{x + 1}$$ **Final answer:** $$\boxed{\frac{2(x^2 + 1)}{x + 1}}$$ --- 6. **Problem:** Simplify $$\left(\frac{x^2 + 6x + 5}{x}\right) \times \left(\frac{x^4}{3x + 3}\right)$$ 7. Factor: - $x^2 + 6x + 5 = (x + 1)(x + 5)$ - $3x + 3 = 3(x + 1)$ 8. Multiply: $$\frac{(x + 1)(x + 5)}{x} \times \frac{x^4}{3(x + 1)} = \frac{(x + 1)(x + 5) x^4}{3 x (x + 1)}$$ 9. Cancel $(x + 1)$ and $x$: $$\frac{(x + 5) x^3}{3}$$ **Final answer:** $$\boxed{\frac{x^3 (x + 5)}{3}}$$ --- 7. **Problem:** Simplify $$\left(\frac{x^2 - 1}{(x - 1)^2}\right) \times \left(\frac{x - 1}{x^2 + 2x + 1}\right)$$ 8. Factor: - $x^2 - 1 = (x - 1)(x + 1)$ - $x^2 + 2x + 1 = (x + 1)^2$ 9. Multiply: $$\frac{(x - 1)(x + 1)}{(x - 1)^2} \times \frac{x - 1}{(x + 1)^2} = \frac{(x - 1)(x + 1)(x - 1)}{(x - 1)^2 (x + 1)^2}$$ 10. Simplify numerator and denominator: $$\frac{(x - 1)^2 (x + 1)}{(x - 1)^2 (x + 1)^2} = \frac{1}{x + 1}$$ **Final answer:** $$\boxed{\frac{1}{x + 1}}$$ --- 8. **Problem:** Simplify $$\left(\frac{x^2 - 2x}{xy - 2y + 3x - 6}\right) \times \left(\frac{8y + 24}{3x + 6}\right)$$ 9. Factor: - $x^2 - 2x = x(x - 2)$ - $xy - 2y + 3x - 6 = y(x - 2) + 3(x - 2) = (x - 2)(y + 3)$ - $8y + 24 = 8(y + 3)$ - $3x + 6 = 3(x + 2)$ 10. Multiply: $$\frac{x(x - 2)}{(x - 2)(y + 3)} \times \frac{8(y + 3)}{3(x + 2)} = \frac{x(x - 2) 8 (y + 3)}{(x - 2)(y + 3) 3 (x + 2)}$$ 11. Cancel $(x - 2)$ and $(y + 3)$: $$\frac{8x}{3(x + 2)}$$ **Final answer:** $$\boxed{\frac{8x}{3(x + 2)}}$$ --- 9. **Problem:** Simplify $$\left(\frac{x}{x^2 - y^2}\right) \times \left(\frac{x + y}{x^2 + xy}\right)$$ 10. Factor: - $x^2 - y^2 = (x - y)(x + y)$ - $x^2 + xy = x(x + y)$ 11. Multiply: $$\frac{x}{(x - y)(x + y)} \times \frac{x + y}{x(x + y)} = \frac{x (x + y)}{(x - y)(x + y) x (x + y)}$$ 12. Cancel $x$ and $(x + y)$: $$\frac{1}{x - y}$$ **Final answer:** $$\boxed{\frac{1}{x - y}}$$ --- 10. **Problem:** Simplify $$\left(\frac{2x^2 - 9x + 9}{8x - 12}\right) \times \left(\frac{2x}{x^2 - 3x}\right)$$ 11. Factor: - $2x^2 - 9x + 9$ factors as $(2x - 3)(x - 3)$ - $8x - 12 = 4(2x - 3)$ - $x^2 - 3x = x(x - 3)$ 12. Multiply: $$\frac{(2x - 3)(x - 3)}{4(2x - 3)} \times \frac{2x}{x(x - 3)} = \frac{(2x - 3)(x - 3) 2x}{4(2x - 3) x (x - 3)}$$ 13. Cancel $(2x - 3)$, $(x - 3)$, and $x$: $$\frac{2}{4} = \frac{1}{2}$$ **Final answer:** $$\boxed{\frac{1}{2}}$$