1. **State the problem:** Simplify the expression $$\frac{5}{x^2 - 5x - 14} \div \frac{5x + 25}{x^2 - 2x - 35}$$. 2. **Rewrite division as multiplication by reciprocal:** $$\frac{5}{x^2 - 5x - 14} \times \frac{x^2 - 2x - 35}{5x + 25}$$ 3. **Factor all polynomials:** - Factor denominator $x^2 - 5x - 14$: $$x^2 - 5x - 14 = (x - 7)(x + 2)$$ - Factor numerator $x^2 - 2x - 35$: $$x^2 - 2x - 35 = (x - 7)(x + 5)$$ - Factor denominator $5x + 25$: $$5x + 25 = 5(x + 5)$$ 4. **Substitute factored forms:** $$\frac{5}{(x - 7)(x + 2)} \times \frac{(x - 7)(x + 5)}{5(x + 5)}$$ 5. **Multiply numerators and denominators:** $$\frac{5 \times (x - 7)(x + 5)}{(x - 7)(x + 2) \times 5(x + 5)}$$ 6. **Cancel common factors:** $$\frac{\cancel{5} \times \cancel{(x - 7)} \times \cancel{(x + 5)}}{\cancel{(x - 7)} \times (x + 2) \times \cancel{5} \times \cancel{(x + 5)}} = \frac{1}{x + 2}$$ 7. **Final simplified expression:** $$\boxed{\frac{1}{x + 2}}$$ This means the original expression simplifies to $\frac{1}{x + 2}$, provided $x \neq 7, -5, -2$ to avoid division by zero.