1. **State the problem:** Simplify the expression $$\frac{x^2 - 49}{2x^2 - 3x - 5} \div \frac{x^2 - x - 42}{x^2 + 7x + 6}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{x^2 - 49}{2x^2 - 3x - 5} \times \frac{x^2 + 7x + 6}{x^2 - x - 42}$$ 3. **Factor all polynomials:** - $x^2 - 49 = (x - 7)(x + 7)$ (difference of squares) - $2x^2 - 3x - 5 = (2x + 5)(x - 1)$ (factor by grouping or quadratic formula) - $x^2 - x - 42 = (x - 7)(x + 6)$ - $x^2 + 7x + 6 = (x + 1)(x + 6)$ 4. **Substitute factored forms:** $$\frac{(x - 7)(x + 7)}{(2x + 5)(x - 1)} \times \frac{(x + 1)(x + 6)}{(x - 7)(x + 6)}$$ 5. **Cancel common factors:** - $(x - 7)$ cancels - $(x + 6)$ cancels Remaining expression: $$\frac{(x + 7)}{(2x + 5)(x - 1)} \times (x + 1) = \frac{(x + 7)(x + 1)}{(2x + 5)(x - 1)}$$ 6. **Final simplified form:** $$\frac{(x + 7)(x + 1)}{(2x + 5)(x - 1)}$$ 7. **Note:** The expression given $$\frac{2x + 7}{x - 5}$$ does not match the simplified result. **Answer:** $$\boxed{\frac{(x + 7)(x + 1)}{(2x + 5)(x - 1)}}$$