1. **State the problem:** Solve the equation $x^2 - 32 = -4x$ for all values of $x$ by completing the square. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 + 4x - 32 = 0$$ 3. **Complete the square:** To complete the square for $x^2 + 4x$, take half of the coefficient of $x$, which is $4$, half is $2$, then square it: $2^2 = 4$. 4. Add and subtract $4$ inside the equation to keep it balanced: $$x^2 + 4x + 4 - 4 - 32 = 0$$ 5. Group the perfect square trinomial and simplify constants: $$ (x + 2)^2 - 36 = 0$$ 6. Isolate the square term: $$ (x + 2)^2 = 36$$ 7. Take the square root of both sides: $$ x + 2 = \pm \sqrt{36}$$ $$ x + 2 = \pm 6$$ 8. Solve for $x$: - For $x + 2 = 6$, subtract 2: $$ x = 6 - 2 = 4$$ - For $x + 2 = -6$, subtract 2: $$ x = -6 - 2 = -8$$ **Final answer:** $$\boxed{x = 4 \text{ or } x = -8}$$