1. The problem asks to find the value of $p^2 + q^2$ given $p = 5 + 2\sqrt{6}$. However, the value of $q$ is not provided in the problem statement, so we cannot directly compute $p^2 + q^2$ without $q$. 2. Since $q$ is missing, let's analyze the given options and the expression for $p$ to see if we can infer $q$ or the value of $p^2 + q^2$. 3. Calculate $p^2$: $$p^2 = (5 + 2\sqrt{6})^2 = 5^2 + 2 \times 5 \times 2\sqrt{6} + (2\sqrt{6})^2 = 25 + 20\sqrt{6} + 4 \times 6 = 25 + 20\sqrt{6} + 24 = 49 + 20\sqrt{6}$$ 4. Without $q$, we cannot find $q^2$ or $p^2 + q^2$. The problem as stated is incomplete. Therefore, the problem cannot be solved with the given information.