1. **State the problem:** Simplify the rational expression $ \frac{4y(y - 3)(y + 4)}{y(y^2 - y - 6)} $. 2. **Factor the denominator:** The quadratic $ y^2 - y - 6 $ factors as $ (y - 3)(y + 2) $ because $ -3 \times 2 = -6 $ and $ -3 + 2 = -1 $. 3. **Rewrite the expression:** $$ \frac{4y(y - 3)(y + 4)}{y(y - 3)(y + 2)} $$ 4. **Cancel common factors:** Both numerator and denominator have $ y $ and $ (y - 3) $ as factors. $$ \frac{\cancel{y} \cdot \cancel{(y - 3)} \cdot 4(y + 4)}{\cancel{y} \cdot \cancel{(y - 3)} (y + 2)} = \frac{4(y + 4)}{y + 2} $$ 5. **Final simplified expression:** $$ \frac{4(y + 4)}{y + 2} $$ 6. **Important note:** The original expression is undefined when the denominator is zero, so $ y \neq 0, y \neq 3, y \neq -2 $.