1. Problem: Multiply \( \frac{5y^3}{32x} \) and \( \frac{-4}{15x^2y^3} \). Step 1: Multiply numerators and denominators: $$\frac{5y^3 \times (-4)}{32x \times 15x^2y^3} = \frac{-20y^3}{480x^3y^3}$$ 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 \( \frac{9x^2}{y^3} \). Step 1: Multiply numerators and denominators: $$\frac{y^3 \times 9x^2}{8 \times y^3} = \frac{9x^2y^3}{8y^3}$$ Step 2: Cancel \( y^3 \): $$\frac{9x^2}{8}$$ --- 3. Problem: Multiply \( \frac{1}{4x - 3} \) and \( 20x - 15 \). Step 1: Factor \( 20x - 15 = 5(4x - 3) \). Step 2: Multiply: $$\frac{1}{4x - 3} \times 5(4x - 3) = \frac{5(4x - 3)}{4x - 3}$$ Step 3: Cancel \( 4x - 3 \): $$5$$ --- 4. Problem: Multiply \( \frac{x - 6}{2x + 5} \) and \( \frac{2x}{-x + 6} \). Step 1: Note \( -x + 6 = -(x - 6) \). Step 2: Multiply: $$\frac{x - 6}{2x + 5} \times \frac{2x}{-(x - 6)} = \frac{(x - 6)2x}{(2x + 5)(-(x - 6))}$$ Step 3: Cancel \( x - 6 \): $$\frac{2x}{-(2x + 5)} = \frac{-2x}{2x + 5}$$ --- 5. Problem: Multiply \( \frac{x^3 + x}{5} \) and \( \frac{10}{x^2 + x} \). 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 \( \frac{x^4}{3x + 3} \). Step 1: Factor: $$x^2 + 6x + 5 = (x + 5)(x + 1)$$ $$3x + 3 = 3(x + 1)$$ Step 2: Multiply: $$\frac{(x + 5)(x + 1)}{x} \times \frac{x^4}{3(x + 1)} = \frac{(x + 5)(x + 1)x^4}{3x(x + 1)}$$ Step 3: Cancel \( x \) and \( x + 1 \): $$\frac{(x + 5)x^3}{3}$$ --- 7. Problem: Multiply \( \frac{x^2 - 1}{(x - 1)^2} \) and \( \frac{x - 1}{x^2 + 2x + 1} \). Step 1: Factor: $$x^2 - 1 = (x - 1)(x + 1)$$ $$x^2 + 2x + 1 = (x + 1)^2$$ 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 \( \frac{8y + 24}{3x + 6} \). 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: $$\frac{x(x - 2)}{(x - 2)(y + 3)} \times \frac{8(y + 3)}{3(x + 2)} = \frac{x(x - 2)8(y + 3)}{(x - 2)(y + 3)3(x + 2)}$$ Step 3: Cancel \( x - 2 \) and \( y + 3 \): $$\frac{8x}{3(x + 2)}$$ --- 9. Problem: Multiply \( \frac{x}{x^2 - y^2} \) and \( \frac{x + y}{x^2 + xy} \). Step 1: Factor denominators: $$x^2 - y^2 = (x - y)(x + y)$$ $$x^2 + xy = x(x + y)$$ Step 2: Multiply: $$\frac{x}{(x - y)(x + y)} \times \frac{x + y}{x(x + y)} = \frac{x(x + y)}{(x - y)(x + y) x (x + y)}$$ Step 3: Cancel \( x \) and \( x + y \): $$\frac{1}{x - y}$$ --- 10. Problem: Multiply \( \frac{2x^2 - 9x + 9}{8x - 12} \) and \( \frac{2x}{x^2 - 3x} \). 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: $$\frac{(2x - 3)(x - 3)}{4(2x - 3)} \times \frac{2x}{x(x - 3)} = \frac{(2x - 3)(x - 3) 2x}{4(2x - 3) x (x - 3)}$$ Step 3: Cancel \( 2x - 3 \), \( x - 3 \), and \( x \): $$\frac{2}{4} = \frac{1}{2}$$