1. **Problem statement:** Simplify the expression $$\frac{6}{2 - \sqrt{10}} + \frac{5\sqrt{2} - 2\sqrt{5}}{\sqrt{5} - \sqrt{2}} - \sqrt{7 - 2\sqrt{10}}$$. 2. **Rationalize denominators:** - For $$\frac{6}{2 - \sqrt{10}}$$, multiply numerator and denominator by the conjugate $$2 + \sqrt{10}$$: $$\frac{6}{2 - \sqrt{10}} \times \frac{2 + \sqrt{10}}{2 + \sqrt{10}} = \frac{6(2 + \sqrt{10})}{(2)^2 - (\sqrt{10})^2} = \frac{6(2 + \sqrt{10})}{4 - 10} = \frac{6(2 + \sqrt{10})}{-6} = - (2 + \sqrt{10})$$. 3. **Simplify the second fraction:** $$\frac{5\sqrt{2} - 2\sqrt{5}}{\sqrt{5} - \sqrt{2}}$$. Multiply numerator and denominator by the conjugate $$\sqrt{5} + \sqrt{2}$$: $$\frac{(5\sqrt{2} - 2\sqrt{5})(\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})} = \frac{(5\sqrt{2})(\sqrt{5}) + (5\sqrt{2})(\sqrt{2}) - (2\sqrt{5})(\sqrt{5}) - (2\sqrt{5})(\sqrt{2})}{5 - 2}$$ Calculate numerator terms: - $$5\sqrt{2} \times \sqrt{5} = 5\sqrt{10}$$ - $$5\sqrt{2} \times \sqrt{2} = 5 \times 2 = 10$$ - $$2\sqrt{5} \times \sqrt{5} = 2 \times 5 = 10$$ - $$2\sqrt{5} \times \sqrt{2} = 2\sqrt{10}$$ So numerator is: $$5\sqrt{10} + 10 - 10 - 2\sqrt{10} = (5\sqrt{10} - 2\sqrt{10}) + (10 - 10) = 3\sqrt{10} + 0 = 3\sqrt{10}$$ Denominator is $$5 - 2 = 3$$. Therefore, the fraction simplifies to: $$\frac{3\sqrt{10}}{3} = \sqrt{10}$$. 4. **Simplify the radical term:** $$\sqrt{7 - 2\sqrt{10}}$$. Try to express inside the root as a perfect square: Assume $$\sqrt{7 - 2\sqrt{10}} = \sqrt{a} - \sqrt{b}$$ for some positive $$a,b$$. Then: $$7 - 2\sqrt{10} = (\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab}$$. Matching terms: $$a + b = 7$$ $$2\sqrt{ab} = 2\sqrt{10} \Rightarrow \sqrt{ab} = \sqrt{10} \Rightarrow ab = 10$$. Solve system: $$a + b = 7$$ $$ab = 10$$ Try factors of 10 that sum to 7: 5 and 2. So $$a=5$$, $$b=2$$. Therefore: $$\sqrt{7 - 2\sqrt{10}} = \sqrt{5} - \sqrt{2}$$. 5. **Combine all parts:** $$- (2 + \sqrt{10}) + \sqrt{10} - (\sqrt{5} - \sqrt{2}) = -2 - \sqrt{10} + \sqrt{10} - \sqrt{5} + \sqrt{2}$$ Simplify terms: $$- \sqrt{10} + \sqrt{10} = 0$$ Final expression: $$-2 - \sqrt{5} + \sqrt{2}$$. **Final answer:** $$\boxed{-2 - \sqrt{5} + \sqrt{2}}$$.