The problem is to simplify the expression $\frac{2\sqrt{5}}{\sqrt{2} - \sqrt{3}}$. We need to rationalize the denominator.

1. Problem Description

The problem is to simplify the expression $\frac{2\sqrt{5}}{\sqrt{2} - \sqrt{3}}$. We need to rationalize the denominator.

2. Solution Steps

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{2} + \sqrt{3}$.

$\frac{2\sqrt{5}}{\sqrt{2} - \sqrt{3}} = \frac{2\sqrt{5}}{\sqrt{2} - \sqrt{3}} \cdot \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}}$

Using the difference of squares formula, $(a - b)(a + b) = a^2 - b^2$, the denominator becomes:

$(\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1$

The numerator becomes:

$2\sqrt{5}(\sqrt{2} + \sqrt{3}) = 2\sqrt{5}\sqrt{2} + 2\sqrt{5}\sqrt{3} = 2\sqrt{10} + 2\sqrt{15}$

So the entire expression is:

$\frac{2\sqrt{10} + 2\sqrt{15}}{-1} = -2\sqrt{10} - 2\sqrt{15}$

3. Final Answer

$-2\sqrt{10} - 2\sqrt{15}$