1. **State the problem:** Solve the equation $4z^2 + 16z = 0$ for $z$. 2. **Formula and rules:** To solve quadratic equations, we can factor the expression and use the zero product property which states if $ab=0$, then either $a=0$ or $b=0$. 3. **Factor the equation:** $$4z^2 + 16z = 0$$ Factor out the common factor $4z$: $$4z(z + 4) = 0$$ 4. **Apply zero product property:** Set each factor equal to zero: $$4z = 0 \quad \text{or} \quad z + 4 = 0$$ 5. **Solve each equation:** $$z = 0$$ $$z = -4$$ 6. **Final answer:** The solutions to the equation are $z = 0$ and $z = -4$.