1. **State the problem:** Solve the equation $-4x^3 - 4x = 0$ for $x$. 2. **Factor the equation:** Factor out the common factor $-4x$: $$-4x^3 - 4x = -4x(x^2 + 1) = 0$$ 3. **Apply the zero product property:** For the product to be zero, either factor must be zero: $$-4x = 0 \quad \text{or} \quad x^2 + 1 = 0$$ 4. **Solve each equation:** - From $-4x = 0$, dividing both sides by $-4$: $$\cancel{-4}x = \cancel{0} \Rightarrow x = 0$$ - From $x^2 + 1 = 0$: $$x^2 = -1$$ Since $x^2$ cannot be negative for real numbers, there are no real solutions here. 5. **Final answer:** The only real solution is: $$x = 0$$