1. **State the problem:** Simplify the expression $$\frac{x^2 + 13x + 12}{x + 1} \times \frac{5x + 10}{4x^2 - 16}$$. 2. **Factor all polynomials:** - Numerator of first fraction: $$x^2 + 13x + 12 = (x + 1)(x + 12)$$ - Denominator of first fraction: $$x + 1$$ (already factored) - Numerator of second fraction: $$5x + 10 = 5(x + 2)$$ - Denominator of second fraction: $$4x^2 - 16 = 4(x^2 - 4) = 4(x - 2)(x + 2)$$ 3. **Rewrite the expression with factors:** $$\frac{(x + 1)(x + 12)}{x + 1} \times \frac{5(x + 2)}{4(x - 2)(x + 2)}$$ 4. **Cancel common factors:** - Cancel $$x + 1$$ from numerator and denominator: $$\frac{\cancel{(x + 1)}(x + 12)}{\cancel{x + 1}} \times \frac{5(x + 2)}{4(x - 2)(x + 2)}$$ - Cancel $$x + 2$$ from numerator and denominator: $$\frac{(x + 12)}{1} \times \frac{5\cancel{(x + 2)}}{4(x - 2)\cancel{(x + 2)}}$$ 5. **Multiply remaining factors:** $$\frac{(x + 12) \times 5}{4(x - 2)} = \frac{5(x + 12)}{4(x - 2)}$$ 6. **Final simplified expression:** $$\boxed{\frac{5(x + 12)}{4(x - 2)}}$$ This is the simplified form of the original expression, valid for all $$x \neq -1, -2, 2$$ to avoid division by zero.