1. **State the problem:** Simplify or factor the cubic polynomial $x^3 + 2x^2 - 5x - 6$. 2. **Formula and rules:** To factor a cubic polynomial, try to find rational roots using the Rational Root Theorem, then use polynomial division or synthetic division to factor. 3. **Find possible roots:** Possible rational roots are factors of the constant term $-6$: $\pm1, \pm2, \pm3, \pm6$. 4. **Test roots:** Substitute $x=1$: $$1^3 + 2(1)^2 - 5(1) - 6 = 1 + 2 - 5 - 6 = -8 \neq 0$$ Substitute $x=2$: $$2^3 + 2(2)^2 - 5(2) - 6 = 8 + 8 - 10 - 6 = 0$$ So, $x=2$ is a root. 5. **Divide polynomial by $(x-2)$:** Using synthetic division: \begin{align*} &2 | 1 \quad 2 \quad -5 \quad -6 \\ &\quad \quad 2 \quad 8 \quad 6 \\ &\quad 1 \quad 4 \quad 3 \quad 0 \end{align*} The quotient is $x^2 + 4x + 3$. 6. **Factor the quadratic:** $$x^2 + 4x + 3 = (x + 1)(x + 3)$$ 7. **Final factorization:** $$x^3 + 2x^2 - 5x - 6 = (x - 2)(x + 1)(x + 3)$$ **Answer:** The polynomial factors as $(x - 2)(x + 1)(x + 3)$.