1. **State the problem:** Show that $x+2$ is a factor of $h(x) = x^3 - x^2 - 24x - 36$ and then factor $h(x)$ completely. 2. **Check if $x+2$ is a factor using the Factor Theorem:** Substitute $x = -2$ into $h(x)$. $$h(-2) = (-2)^3 - (-2)^2 - 24(-2) - 36 = -8 - 4 + 48 - 36 = 0$$ Since $h(-2) = 0$, $x+2$ is a factor. 3. **Divide $h(x)$ by $x+2$ using synthetic division:** Coefficients: 1 (for $x^3$), -1 (for $x^2$), -24 (for $x$), -36 (constant) -2 | 1 -1 -24 -36 | -2 6 36 ---------------- 1 -3 -18 0 The quotient is $x^2 - 3x - 18$. 4. **Factor the quadratic $x^2 - 3x - 18$:** Find two numbers that multiply to $-18$ and add to $-3$: these are $-6$ and $3$. So, $$x^2 - 3x - 18 = (x - 6)(x + 3)$$ 5. **Write the complete factorization:** $$h(x) = (x + 2)(x - 6)(x + 3)$$ **Final answer:** $h(x) = (x + 2)(x - 6)(x + 3)$