The given expression to simplify is:

$\frac{5\sqrt{2}}{\sqrt{10} - \sqrt{5}} - \frac{3\sqrt{5}}{\sqrt{5} - \sqrt{2}} - \frac{\sqrt{10}}{\sqrt{2} - 1}$

To simplify this, we will rationalize each denominator and work through the terms step by step. Let me start by simplifying the terms.

Step 1: Rationalize the first term

$\frac{5\sqrt{2}}{\sqrt{10} - \sqrt{5}}$

We multiply both the numerator and the denominator by the conjugate of the denominator: $\frac{5\sqrt{2}}{(\sqrt{10} - \sqrt{5})} \cdot \frac{(\sqrt{10} + \sqrt{5})}{(\sqrt{10} + \sqrt{5})} = \frac{5\sqrt{2}(\sqrt{10} + \sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}$

Simplifying the denominator: $(\sqrt{10})^2 - (\sqrt{5})^2 = 10 - 5 = 5$

Now, simplify the numerator: $5\sqrt{2}(\sqrt{10} + \sqrt{5}) = 5\sqrt{20} + 5\sqrt{10} = 10\sqrt{5} + 5\sqrt{10}$

Thus, the first term becomes: $\frac{10\sqrt{5} + 5\sqrt{10}}{5} = 2\sqrt{5} + \sqrt{10}$

Step 2: Rationalize the second term

$\frac{3\sqrt{5}}{\sqrt{5} - \sqrt{2}}$

Multiply by the conjugate of the denominator: $\frac{3\sqrt{5}}{(\sqrt{5} - \sqrt{2})} \cdot \frac{(\sqrt{5} + \sqrt{2})}{(\sqrt{5} + \sqrt{2})} = \frac{3\sqrt{5}(\sqrt{5} + \sqrt{2})}{(\sqrt{5})^2 - (\sqrt{2})^2}$

Simplifying the denominator: $(\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3$

Simplify the numerator: $3\sqrt{5}(\sqrt{5} + \sqrt{2}) = 3 \times 5 + 3\sqrt{10} = 15 + 3\sqrt{10}$

Thus, the second term becomes: $\frac{15 + 3\sqrt{10}}{3} = 5 + \sqrt{10}$

Step 3: Rationalize the third term

$\frac{\sqrt{10}}{\sqrt{2} - 1}$

Multiply by the conjugate of the denominator: $\frac{\sqrt{10}}{(\sqrt{2} - 1)} \cdot \frac{(\sqrt{2} + 1)}{(\sqrt{2} + 1)} = \frac{\sqrt{10}(\sqrt{2} + 1)}{(\sqrt{2})^2 - (1)^2}$

Simplifying the denominator: $(\sqrt{2})^2 - 1^2 = 2 - 1 = 1$

Thus, the third term becomes: $\sqrt{10}(\sqrt{2} + 1) = \sqrt{20} + \sqrt{10} = 2\sqrt{5} + \sqrt{10}$

Final Expression

Now, summing up all three terms: $(2\sqrt{5} + \sqrt{10}) - (5 + \sqrt{10}) - (2\sqrt{5} + \sqrt{10})$

Simplifying the expression: $2\sqrt{5} + \sqrt{10} - 5 - \sqrt{10} - 2\sqrt{5} - \sqrt{10}$

The terms $2\sqrt{5}$ and $-2\sqrt{5}$ cancel out, and $\sqrt{10} - \sqrt{10} - \sqrt{10}$ simplifies to $-\sqrt{10}$. So we are left with: $-5 - \sqrt{10}$

Thus, the simplified expression is: $\boxed{-5 - \sqrt{10}}$

Would you like more details or clarifications on any part of the process?

Relative Questions:

1. How do you rationalize a denominator involving square roots?
2. What are conjugates, and how do they help simplify expressions?
3. Why do the terms cancel out in the final step of simplification?
4. How does squaring square roots help in rationalizing denominators?
5. Can this method be applied to more complex radical expressions?

Tip: Always simplify square roots and perform factorization if possible before combining terms to make the process easier.