1. **State the problem:** Factor the cubic polynomial $x^3 + x^2 - 4x - 4$ using the factor theorem. 2. **Recall the factor theorem:** If $f(a) = 0$ for some number $a$, then $(x - a)$ is a factor of $f(x)$. 3. **Find possible rational roots:** Possible roots are factors of the constant term $-4$, i.e., $\pm1, \pm2, \pm4$. 4. **Test $x=1$:** $$f(1) = 1^3 + 1^2 - 4(1) - 4 = 1 + 1 - 4 - 4 = -6 \neq 0$$ 5. **Test $x=-1$:** $$f(-1) = (-1)^3 + (-1)^2 - 4(-1) - 4 = -1 + 1 + 4 - 4 = 0$$ Since $f(-1) = 0$, $(x + 1)$ is a factor. 6. **Divide $f(x)$ by $(x + 1)$ using polynomial division:** $$\frac{x^3 + x^2 - 4x - 4}{x + 1} = x^2 \cancel{+ x} - 2x \cancel{- 2}$$ 7. **Write the quotient:** $$x^2 - 2x - 4$$ 8. **Factor the quadratic $x^2 - 2x - 4$ using the quadratic formula:** $$x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-4)}}{2(1)} = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5}$$ 9. **Final factorization:** $$f(x) = (x + 1)(x - (1 + \sqrt{5}))(x - (1 - \sqrt{5}))$$ This is the complete factorization using the factor theorem and quadratic formula.