1. Stating the problem: We need to find the simplified form of the quotient of two expressions: $$\frac{x^2 - 5x}{x + 2} \div \frac{3x - 15}{x^2 - 4}$$ where $x \neq -2, 2, 5$. 2. Formula and rules: Dividing by a fraction is equivalent to multiplying by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 3. Apply this to our problem: $$\frac{x^2 - 5x}{x + 2} \times \frac{x^2 - 4}{3x - 15}$$ 4. Factor all polynomials: - $x^2 - 5x = x(x - 5)$ - $x^2 - 4 = (x - 2)(x + 2)$ (difference of squares) - $3x - 15 = 3(x - 5)$ 5. Substitute factored forms: $$\frac{x(x - 5)}{x + 2} \times \frac{(x - 2)(x + 2)}{3(x - 5)}$$ 6. Simplify by canceling common factors: - Cancel $(x + 2)$ in numerator and denominator - Cancel $(x - 5)$ in numerator and denominator Resulting expression: $$\frac{x}{1} \times \frac{x - 2}{3} = \frac{x(x - 2)}{3}$$ 7. Final answer: $$\boxed{\frac{x(x - 2)}{3}}$$ This corresponds to option A.