1. **Problem:** Given $p = 5 + 2\sqrt{6}$, find the value of $p^2 + p - 1$. 2. **Formula and rules:** To find $p^2 + p - 1$, first calculate $p^2$, then add $p$, and subtract 1. 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. **Calculate $p^2 + p - 1$:** $$p^2 + p - 1 = (49 + 20\sqrt{6}) + (5 + 2\sqrt{6}) - 1 = (49 + 5 - 1) + (20\sqrt{6} + 2\sqrt{6}) = 53 + 22\sqrt{6}$$ 5. **Simplify and interpret:** The expression $53 + 22\sqrt{6}$ is the exact value. Since the problem provides options (7, 10, 21, 3), and none match this expression, check if the problem expects a numerical approximation or if the problem is different. However, since the user asked for $p^2 + p - 1$ with $p=5+2\sqrt{6}$, this is the exact value. **Final answer:** $$p^2 + p - 1 = 53 + 22\sqrt{6}$$