6. (√6 - 6)² + (9 + √6)² =

$$(\sqrt{6} - 6)^2 + (9 + \sqrt{6})^2 = (\sqrt{6})^2 - 2 \cdot 6\sqrt{6} + 6^2 + 9^2 + 2 \cdot 9\sqrt{6} + (\sqrt{6})^2 = 6 - 12\sqrt{6} + 36 + 81 + 18\sqrt{6} + 6 = 129 + 6\sqrt{6}$$