1. **State the problem:** Factorize the cubic polynomial $$x^3 - 3x^2 - 4x + 12$$. 2. **Recall the factoring approach:** To factor a cubic polynomial, try to find rational roots using the Rational Root Theorem, then use polynomial division or synthetic division to factor out the root. 3. **Find possible roots:** Possible rational roots are factors of the constant term 12: $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm12$$. 4. **Test roots:** Substitute these values into the polynomial to find which make it zero. - For $$x=3$$: $$3^3 - 3\times3^2 - 4\times3 + 12 = 27 - 27 - 12 + 12 = 0$$ So, $$x=3$$ is a root. 5. **Divide polynomial by $$x-3$$:** Using polynomial division or synthetic division: $$\frac{x^3 - 3x^2 - 4x + 12}{x - 3} = x^2 - 4$$ 6. **Factor the quotient:** $$x^2 - 4$$ is a difference of squares: $$x^2 - 4 = (x - 2)(x + 2)$$ 7. **Write the full factorization:** $$x^3 - 3x^2 - 4x + 12 = (x - 3)(x - 2)(x + 2)$$ **Final answer:** $$\boxed{(x - 3)(x - 2)(x + 2)}$$