1. **Problem:** Factor the trinomial $x^2 + 12x + 36$ completely. 2. **Formula and rules:** A trinomial of the form $x^2 + bx + c$ can be factored as $(x + m)(x + n)$ where $m$ and $n$ satisfy $m + n = b$ and $mn = c$. 3. **Step-by-step solution:** - Identify $b = 12$ and $c = 36$. - Find two numbers $m$ and $n$ such that $m + n = 12$ and $mn = 36$. - The numbers $6$ and $6$ satisfy these conditions because $6 + 6 = 12$ and $6 \times 6 = 36$. - Therefore, the factorization is: $$x^2 + 12x + 36 = (x + 6)(x + 6) = (x + 6)^2$$ 4. **Explanation:** This trinomial is a perfect square trinomial because it can be written as the square of a binomial. 5. **Final answer:** $$\boxed{(x + 6)^2}$$