1. Problem: Simplify the product \( \frac{5y^3}{32x} \cdot \frac{-4}{15x^2 y^3} \). Step 1: Multiply numerators and denominators: $$\frac{5y^3 \cdot (-4)}{32x \cdot 15x^2 y^3} = \frac{-20y^3}{480x^3 y^3}$$ 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: Simplify \( \frac{y^3}{8} \cdot \frac{9x^2}{y^3} \). Step 1: Multiply numerators and denominators: $$\frac{y^3 \cdot 9x^2}{8 \cdot y^3} = \frac{9x^2 y^3}{8 y^3}$$ Step 2: Cancel \(y^3\): $$\frac{9x^2}{8}$$ 3. Problem: Simplify \( \frac{1}{4x - 3} \cdot (20x - 15) \). Step 1: Factor \(20x - 15 = 5(4x - 3)\). Step 2: Multiply: $$\frac{1}{4x - 3} \cdot 5(4x - 3) = 5$$ 4. Problem: Simplify \( \frac{x - 6}{2x + 5} \cdot \frac{2x}{-x + 6} \). Step 1: Note \(-x + 6 = -(x - 6)\). Step 2: Multiply: $$\frac{x - 6}{2x + 5} \cdot \frac{2x}{-(x - 6)} = \frac{(x - 6) 2x}{(2x + 5)(-(x - 6))} = \frac{2x}{-(2x + 5)} = -\frac{2x}{2x + 5}$$ 5. Problem: Simplify \( \frac{x^3 + x}{5} \cdot \frac{10}{x^2 + x} \). Step 1: Factor numerator and denominator: $$x^3 + x = x(x^2 + 1), \quad x^2 + x = x(x + 1)$$ Step 2: Multiply: $$\frac{x(x^2 + 1)}{5} \cdot \frac{10}{x(x + 1)} = \frac{10x(x^2 + 1)}{5x(x + 1)}$$ Step 3: Cancel \(x\) and simplify \(\frac{10}{5} = 2\): $$\frac{2(x^2 + 1)}{x + 1}$$ 6. Problem: Simplify \( \frac{x^2 + 6x + 5}{x} \cdot \frac{x^4}{3x + 3} \). Step 1: Factor: $$x^2 + 6x + 5 = (x + 1)(x + 5), \quad 3x + 3 = 3(x + 1)$$ Step 2: Multiply: $$\frac{(x + 1)(x + 5)}{x} \cdot \frac{x^4}{3(x + 1)} = \frac{(x + 1)(x + 5) x^4}{x 3 (x + 1)}$$ Step 3: Cancel \(x + 1\) and \(x\): $$\frac{(x + 5) x^3}{3}$$ 7. Problem: Simplify \( \frac{x^2 - 1}{(x - 1)^2} \cdot \frac{x - 1}{x^2 + 2x + 1} \). Step 1: Factor: $$x^2 - 1 = (x - 1)(x + 1), \quad x^2 + 2x + 1 = (x + 1)^2$$ Step 2: Multiply: $$\frac{(x - 1)(x + 1)}{(x - 1)^2} \cdot \frac{x - 1}{(x + 1)^2} = \frac{(x - 1)(x + 1)(x - 1)}{(x - 1)^2 (x + 1)^2}$$ Step 3: Cancel \((x - 1)^2\): $$\frac{x + 1}{(x + 1)^2} = \frac{1}{x + 1}$$ 8. Problem: Simplify \( \frac{x^2 - 2x}{xy - 2y + 3x - 6} \cdot \frac{8y + 24}{3x + 6} \). Step 1: Factor denominators and numerators: $$x^2 - 2x = x(x - 2)$$ $$xy - 2y + 3x - 6 = y(x - 2) + 3(x - 2) = (x - 2)(y + 3)$$ $$8y + 24 = 8(y + 3)$$ $$3x + 6 = 3(x + 2)$$ Step 2: Multiply: $$\frac{x(x - 2)}{(x - 2)(y + 3)} \cdot \frac{8(y + 3)}{3(x + 2)} = \frac{x \cancel{(x - 2)} 8 \cancel{(y + 3)}}{\cancel{(x - 2)} (y + 3) 3 (x + 2)} = \frac{8x}{3(x + 2)}$$ 9. Problem: Simplify \( \frac{x}{x^2 - y^2} \cdot \frac{x + y}{x^2 + xy} \). Step 1: Factor denominators: $$x^2 - y^2 = (x - y)(x + y)$$ $$x^2 + xy = x(x + y)$$ Step 2: Multiply: $$\frac{x}{(x - y)(x + y)} \cdot \frac{x + y}{x(x + y)} = \frac{x (x + y)}{(x - y)(x + y) x (x + y)}$$ Step 3: Cancel \(x\) and \(x + y\): $$\frac{1}{x - y}$$ 10. Problem: Simplify \( \frac{2x^2 - 9x + 9}{8x - 12} \cdot \frac{2x}{x^2 - 3x} \). Step 1: Factor: $$2x^2 - 9x + 9 = (2x - 3)(x - 3)$$ $$8x - 12 = 4(2x - 3)$$ $$x^2 - 3x = x(x - 3)$$ Step 2: Multiply: $$\frac{(2x - 3)(x - 3)}{4(2x - 3)} \cdot \frac{2x}{x(x - 3)} = \frac{(2x - 3)(x - 3) 2x}{4(2x - 3) x (x - 3)}$$ Step 3: Cancel \(2x - 3\), \(x - 3\), and \(x\): $$\frac{2}{4} = \frac{1}{2}$$ Final answers: 1. \( -\frac{1}{24x^3} \) 2. \( \frac{9x^2}{8} \) 3. \( 5 \) 4. \( -\frac{2x}{2x + 5} \) 5. \( \frac{2(x^2 + 1)}{x + 1} \) 6. \( \frac{(x + 5) x^3}{3} \) 7. \( \frac{1}{x + 1} \) 8. \( \frac{8x}{3(x + 2)} \) 9. \( \frac{1}{x - y} \) 10. \( \frac{1}{2} \)