1. **State the problem:** Simplify the expression $$\frac{x^2 - 25}{x^2 + 5x} \div \frac{xy + 6x - 5y - 30}{5x - 15}$$. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$. 3. **Apply the rule:** $$\frac{x^2 - 25}{x^2 + 5x} \times \frac{5x - 15}{xy + 6x - 5y - 30}$$. 4. **Factor all polynomials:** - Numerator 1: $$x^2 - 25 = (x - 5)(x + 5)$$ (difference of squares). - Denominator 1: $$x^2 + 5x = x(x + 5)$$ (factor out common x). - Numerator 2: $$5x - 15 = 5(x - 3)$$ (factor out 5). - Denominator 2: $$xy + 6x - 5y - 30$$ group terms: $$x(y + 6) - 5(y + 6) = (x - 5)(y + 6)$$. 5. **Rewrite the expression with factors:** $$\frac{(x - 5)(x + 5)}{x(x + 5)} \times \frac{5(x - 3)}{(x - 5)(y + 6)}$$. 6. **Multiply the fractions:** $$\frac{(x - 5)(x + 5) \times 5(x - 3)}{x(x + 5)(x - 5)(y + 6)}$$. 7. **Cancel common factors:** - Cancel $(x - 5)$ numerator and denominator. - Cancel $(x + 5)$ numerator and denominator. Intermediate step with cancellation: $$\frac{\cancel{(x - 5)} \cancel{(x + 5)} \times 5(x - 3)}{x \cancel{(x + 5)} \cancel{(x - 5)} (y + 6)} = \frac{5(x - 3)}{x(y + 6)}$$. 8. **Final simplified expression:** $$\boxed{\frac{5(x - 3)}{x(y + 6)}}$$. This is the simplified form of the original expression.