1. Problem: Multiply $\frac{5y^3}{32x}$ and $\frac{-4}{15x^2 y^3}$ and simplify. Step 1: Multiply numerators and denominators: $$\frac{5y^3 \times (-4)}{32x \times 15x^2 y^3} = \frac{-20y^3}{480x^3 y^3}$$ Step 2: Cancel common factors $y^3$: $$\frac{-20}{480x^3}$$ Step 3: Simplify $\frac{-20}{480} = \frac{-1}{24}$: $$\frac{-1}{24x^3}$$ 2. Problem: Multiply $\frac{y^3}{8}$ and $\frac{9x^2}{y^3}$ and simplify. Step 1: Multiply numerators and denominators: $$\frac{y^3 \times 9x^2}{8 \times y^3} = \frac{9x^2 y^3}{8 y^3}$$ Step 2: Cancel $y^3$: $$\frac{9x^2}{8}$$ 3. Problem: Multiply $\frac{1}{4x - 3}$ and $(20x - 15)$ and simplify. Step 1: Factor numerator of second term: $$20x - 15 = 5(4x - 3)$$ Step 2: Multiply: $$\frac{1}{4x - 3} \times 5(4x - 3) = 5 \times \frac{4x - 3}{4x - 3} = 5$$ 4. Problem: Multiply $\frac{x - 6}{2x + 5}$ and $\frac{2x}{-x + 6}$ and simplify. Step 1: Note $-x + 6 = -(x - 6)$ Step 2: Multiply: $$\frac{x - 6}{2x + 5} \times \frac{2x}{-(x - 6)} = \frac{(x - 6) 2x}{(2x + 5)(-(x - 6))} = \frac{2x (x - 6)}{-(2x + 5)(x - 6)}$$ Step 3: Cancel $x - 6$: $$\frac{2x}{-(2x + 5)} = -\frac{2x}{2x + 5}$$ 5. Problem: Multiply $\frac{x^3 + x}{5}$ and $\frac{10}{x^2 + x}$ and simplify. Step 1: Factor numerator and denominator: $$x^3 + x = x(x^2 + 1)$$ $$x^2 + x = x(x + 1)$$ Step 2: Multiply: $$\frac{x(x^2 + 1)}{5} \times \frac{10}{x(x + 1)} = \frac{10x(x^2 + 1)}{5x(x + 1)}$$ Step 3: Cancel $x$ and simplify $\frac{10}{5} = 2$: $$\frac{2(x^2 + 1)}{x + 1}$$ 6. Problem: Multiply $\frac{x^2 + 6x + 5}{x}$ and $\frac{x^4}{3x + 3}$ and simplify. Step 1: Factor numerator and denominator: $$x^2 + 6x + 5 = (x + 1)(x + 5)$$ $$3x + 3 = 3(x + 1)$$ Step 2: Multiply: $$\frac{(x + 1)(x + 5)}{x} \times \frac{x^4}{3(x + 1)} = \frac{(x + 1)(x + 5) x^4}{x 3 (x + 1)}$$ Step 3: Cancel $x + 1$ and $x$: $$\frac{(x + 5) x^3}{3}$$ 7. Problem: Multiply $\frac{x^2 - 1}{(x - 1)^2}$ and $\frac{x - 1}{x^2 + 2x + 1}$ and simplify. Step 1: Factor expressions: $$x^2 - 1 = (x - 1)(x + 1)$$ $$x^2 + 2x + 1 = (x + 1)^2$$ Step 2: Multiply: $$\frac{(x - 1)(x + 1)}{(x - 1)^2} \times \frac{x - 1}{(x + 1)^2} = \frac{(x - 1)(x + 1)(x - 1)}{(x - 1)^2 (x + 1)^2}$$ Step 3: Simplify numerator and denominator: $$\frac{(x - 1)^2 (x + 1)}{(x - 1)^2 (x + 1)^2} = \frac{1}{x + 1}$$ 8. Problem: Multiply $\frac{x^2 - 2x}{xy - 2y + 3x - 6}$ and $\frac{8y + 24}{3x + 6}$ and simplify. Step 1: Factor all expressions: $$x^2 - 2x = x(x - 2)$$ $$xy - 2y + 3x - 6 = y(x - 2) + 3(x - 2) = (x - 2)(y + 3)$$ $$8y + 24 = 8(y + 3)$$ $$3x + 6 = 3(x + 2)$$ Step 2: Multiply: $$\frac{x(x - 2)}{(x - 2)(y + 3)} \times \frac{8(y + 3)}{3(x + 2)} = \frac{x \cancel{(x - 2)} 8 \cancel{(y + 3)}}{\cancel{(x - 2)} (y + 3) 3 (x + 2)} = \frac{8x}{3(x + 2)}$$ 9. Problem: Multiply $\frac{x}{x^2 - y^2}$ and $\frac{x + y}{x^2 + xy}$ and simplify. Step 1: Factor denominators: $$x^2 - y^2 = (x - y)(x + y)$$ $$x^2 + xy = x(x + y)$$ Step 2: Multiply: $$\frac{x}{(x - y)(x + y)} \times \frac{x + y}{x(x + y)} = \frac{x (x + y)}{(x - y)(x + y) x (x + y)}$$ Step 3: Cancel $x$, $x + y$: $$\frac{1}{x - y}$$ 10. Problem: Multiply $\frac{2x^2 - 9x + 9}{8x - 12}$ and $\frac{2x}{x^2 - 3x}$ and simplify. Step 1: Factor expressions: $$2x^2 - 9x + 9 = (2x - 3)(x - 3)$$ $$8x - 12 = 4(2x - 3)$$ $$x^2 - 3x = x(x - 3)$$ Step 2: Multiply: $$\frac{(2x - 3)(x - 3)}{4(2x - 3)} \times \frac{2x}{x(x - 3)} = \frac{(2x - 3)(x - 3) 2x}{4(2x - 3) x (x - 3)}$$ Step 3: Cancel $(2x - 3)$, $(x - 3)$, and $x$: $$\frac{2}{4} = \frac{1}{2}$$ Final answers: 1. $\boxed{-\frac{1}{24x^3}}$ 2. $\boxed{\frac{9x^2}{8}}$ 3. $\boxed{5}$ 4. $\boxed{-\frac{2x}{2x + 5}}$ 5. $\boxed{\frac{2(x^2 + 1)}{x + 1}}$ 6. $\boxed{\frac{(x + 5) x^3}{3}}$ 7. $\boxed{\frac{1}{x + 1}}$ 8. $\boxed{\frac{8x}{3(x + 2)}}$ 9. $\boxed{\frac{1}{x - y}}$ 10. $\boxed{\frac{1}{2}}$