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. 3. **Rewrite the expression:** $$\frac{x^2 - 25}{x^2 + 5x} \times \frac{5x - 15}{xy + 6x - 5y - 30}$$ 4. **Factor all polynomials:** - Numerator top-left: $$x^2 - 25 = (x - 5)(x + 5)$$ (difference of squares) - Denominator top-left: $$x^2 + 5x = x(x + 5)$$ - Numerator top-right: $$5x - 15 = 5(x - 3)$$ - Denominator top-right: $$xy + 6x - 5y - 30$$ group terms: $$x(y + 6) - 5(y + 6) = (x - 5)(y + 6)$$ 5. **Substitute factored forms:** $$\frac{(x - 5)(x + 5)}{x(x + 5)} \times \frac{5(x - 3)}{(x - 5)(y + 6)}$$ 6. **Multiply numerators and denominators:** $$\frac{(x - 5)(x + 5) \times 5(x - 3)}{x(x + 5) \times (x - 5)(y + 6)}$$ 7. **Cancel common factors:** - Cancel $(x - 5)$ numerator and denominator - Cancel $(x + 5)$ numerator and denominator Intermediate step showing cancellation: $$\frac{\cancel{(x - 5)} \cancel{(x + 5)} \times 5(x - 3)}{x \cancel{(x + 5)} \times \cancel{(x - 5)} (y + 6)} = \frac{5(x - 3)}{x(y + 6)}$$ 8. **Final simplified expression:** $$\frac{5(x - 3)}{x(y + 6)}$$ This is the simplified form of the original expression.