1. **Problem:** Simplify and analyze the expressions $(x + 6)^2$ and $x^2 - 36$. 2. **Formula and rules:** - Square of a binomial: $ (a + b)^2 = a^2 + 2ab + b^2 $ - Difference of squares: $ a^2 - b^2 = (a - b)(a + b) $ 3. **Step-by-step solution:** - Expand $(x + 6)^2$ using the binomial square formula: $$ (x + 6)^2 = x^2 + 2 \cdot x \cdot 6 + 6^2 = x^2 + 12x + 36 $$ - Factor $x^2 - 36$ using the difference of squares: $$ x^2 - 36 = (x - 6)(x + 6) $$ 4. **Explanation:** - The first expression expands to a quadratic trinomial. - The second expression factors into two binomials representing the difference of squares. **Final answers:** - $(x + 6)^2 = x^2 + 12x + 36$ - $x^2 - 36 = (x - 6)(x + 6)$