1. **State the problem:** Calculate the value of the expression $$(-2 + 3\sqrt{6})^2$$. 2. **Recall the formula:** The square of a binomial $(a + b)^2$ is given by $$a^2 + 2ab + b^2$$. 3. **Identify terms:** Here, $a = -2$ and $b = 3\sqrt{6}$. 4. **Calculate each term:** - $a^2 = (-2)^2 = 4$ - $2ab = 2 \times (-2) \times 3\sqrt{6} = -12\sqrt{6}$ - $b^2 = (3\sqrt{6})^2 = 9 \times 6 = 54$ 5. **Sum all terms:** $$4 - 12\sqrt{6} + 54 = 58 - 12\sqrt{6}$$ 6. **Final answer:** $$(-2 + 3\sqrt{6})^2 = 58 - 12\sqrt{6}$$