1. **Problem:** Simplify the rational expression $\frac{12w^4 + 30w^3}{16w^2 + 40w}$. 2. **Formula and rules:** To simplify rational expressions, factor numerator and denominator completely and cancel common factors. 3. **Step 1: Factor numerator and denominator.** $$12w^4 + 30w^3 = 6w^3(2w + 5)$$ $$16w^2 + 40w = 8w(2w + 5)$$ 4. **Step 2: Write the expression with factored forms.** $$\frac{6w^3(2w + 5)}{8w(2w + 5)}$$ 5. **Step 3: Cancel common factors $2w + 5$ and $w$.** $$\frac{6\cancel{w^3}(2w + 5)}{8\cancel{w}(2w + 5)} = \frac{6w^2}{8}$$ 6. **Step 4: Simplify the fraction $\frac{6w^2}{8}$.** $$\frac{6w^2}{8} = \frac{3w^2}{4}$$ **Final answer:** $$\boxed{\frac{3w^2}{4}}$$