1. **State the problem:** Factor the polynomial $x^3 - 216$. 2. **Recognize the form:** This is a difference of cubes since $216 = 6^3$. 3. **Formula for difference of cubes:** $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ 4. **Apply the formula:** Let $a = x$ and $b = 6$. $$x^3 - 6^3 = (x - 6)(x^2 + 6x + 36)$$ 5. **Check if the quadratic factor can be factored further:** The quadratic $x^2 + 6x + 36$ has discriminant $\Delta = 6^2 - 4 \times 1 \times 36 = 36 - 144 = -108 < 0$, so it cannot be factored over the reals. 6. **Final answer:** $$x^3 - 216 = (x - 6)(x^2 + 6x + 36)$$