1. **State the problem:** Simplify the rational expression $$\frac{4r^2 + 24r + 32}{16r^2 - 16r - 96}$$. 2. **Factor numerator and denominator:** - Numerator: $$4r^2 + 24r + 32 = 4(r^2 + 6r + 8)$$. - Factor inside parentheses: $$r^2 + 6r + 8 = (r + 2)(r + 4)$$. - So numerator is $$4(r + 2)(r + 4)$$. - Denominator: $$16r^2 - 16r - 96 = 16(r^2 - r - 6)$$. - Factor inside parentheses: $$r^2 - r - 6 = (r - 3)(r + 2)$$. - So denominator is $$16(r - 3)(r + 2)$$. 3. **Rewrite the expression:** $$\frac{4(r + 2)(r + 4)}{16(r - 3)(r + 2)}$$ 4. **Cancel common factors:** The factor $$(r + 2)$$ appears in numerator and denominator, so cancel it: $$\frac{4\cancel{(r + 2)}(r + 4)}{16(r - 3)\cancel{(r + 2)}} = \frac{4(r + 4)}{16(r - 3)}$$ 5. **Simplify the coefficients:** $$\frac{4(r + 4)}{16(r - 3)} = \frac{\cancel{4}(r + 4)}{\cancel{16}(r - 3)} = \frac{r + 4}{4(r - 3)}$$ **Final answer:** $$\frac{r + 4}{4(r - 3)}$$