1. **State the problem:** We need to simplify the product $ (2 + \sqrt{7})(2 + \sqrt{7}) $. 2. **Formula used:** When multiplying two binomials of the form $(a + b)(a + b)$, we use the formula for the square of a binomial: $$ (a + b)^2 = a^2 + 2ab + b^2 $$ 3. **Apply the formula:** Here, $a = 2$ and $b = \sqrt{7}$. $$ (2 + \sqrt{7})^2 = 2^2 + 2 \times 2 \times \sqrt{7} + (\sqrt{7})^2 $$ 4. **Calculate each term:** $$ 2^2 = 4 $$ $$ 2 \times 2 \times \sqrt{7} = 4\sqrt{7} $$ $$ (\sqrt{7})^2 = 7 $$ 5. **Sum all terms:** $$ 4 + 4\sqrt{7} + 7 = (4 + 7) + 4\sqrt{7} = 11 + 4\sqrt{7} $$ 6. **Final answer:** $$ \boxed{11 + 4\sqrt{7}} $$