Rationalize the denominator.

To rationalize the denominator $\sqrt{5} - \sqrt{2}$, multiply the numerator and denominator by its conjugate, which is $\sqrt{5} + \sqrt{2}$.

$$\frac{\sqrt{2} + 2\sqrt{5}}{\sqrt{5} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}}$$

$$(\sqrt{2} + 2\sqrt{5})(\sqrt{5} + \sqrt{2}) = \sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{2} + 2\sqrt{5}\sqrt{5} + 2\sqrt{5}\sqrt{2}$$ $$= \sqrt{10} + 2 + 2(5) + 2\sqrt{10}$$ $$= \sqrt{10} + 2 + 10 + 2\sqrt{10}$$ $$= 12 + 3\sqrt{10}$$

Use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$:

$$(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2$$ $$= 5 - 2$$ $$= 3$$

$$\frac{12}{3} + \frac{3\sqrt{10}}{3}$$ $$= 4 + \sqrt{10}$$

The simplified expression is $4 + \sqrt{10}$.