1. The problem is to simplify and evaluate $$(\sqrt{2} - 4\sqrt{3})^2$$. 2. We use the formula for the square of a binomial: $$(a - b)^2 = a^2 - 2ab + b^2$$. 3. Let $a = \sqrt{2}$ and $b = 4\sqrt{3}$. 4. Calculate each term: - $a^2 = (\sqrt{2})^2 = 2$ - $b^2 = (4\sqrt{3})^2 = 16 \times 3 = 48$ - $-2ab = -2 \times \sqrt{2} \times 4\sqrt{3} = -8 \sqrt{6}$ 5. Substitute back: $$2 - 8\sqrt{6} + 48$$ 6. Combine like terms: $$50 - 8\sqrt{6}$$ 7. So, the simplified form of $$(\sqrt{2} - 4\sqrt{3})^2$$ is $$50 - 8\sqrt{6}$$.