1. **State the problem:** Simplify the expression $$\frac{x^{2} + 2x - 48}{x^{2} + x - 42} \div \frac{x^{2} + 2x - 15}{x^{2} + 15x + 56}$$ given the factorizations: $$x^{2} + 2x - 48 = (x + 8)(x - 6)$$ $$x^{2} + x - 42 = (x + 7)(x - 6)$$ $$x^{2} + 2x - 15 = (x + 5)(x - 3)$$ $$x^{2} + 15x + 56 = (x + 7)(x + 8)$$ 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{(x + 8)(x - 6)}{(x + 7)(x - 6)} \times \frac{(x + 7)(x + 8)}{(x + 5)(x - 3)}$$ 3. **Cancel common factors:** - Cancel $(x - 6)$ from numerator and denominator: $$\frac{(x + 8)\cancel{(x - 6)}}{(x + 7)\cancel{(x - 6)}} \times \frac{(x + 7)(x + 8)}{(x + 5)(x - 3)}$$ - Cancel $(x + 7)$ from numerator and denominator: $$\frac{(x + 8)}{\cancel{(x + 7)}} \times \frac{\cancel{(x + 7)}(x + 8)}{(x + 5)(x - 3)}$$ 4. **Multiply the remaining factors:** $$\frac{(x + 8)}{1} \times \frac{(x + 8)}{(x + 5)(x - 3)} = \frac{(x + 8)(x + 8)}{(x + 5)(x - 3)} = \frac{(x + 8)^2}{(x + 5)(x - 3)}$$ 5. **Final simplified expression:** $$\boxed{\frac{(x + 8)^2}{(x + 5)(x - 3)}}$$ This is the simplified form of the original expression after factoring, rewriting division as multiplication, canceling common factors, and multiplying remaining terms.