Rationalize the denominator.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Rationalize the denominator.
To rationalize the denominator $\sqrt{5} - \sqrt{2}$, multiply the numerator and denominator by its conjugate, which is $\sqrt{5} + \sqrt{2}$.
$$ \frac{\sqrt{2} + 2\sqrt{5}}{\sqrt{5} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}} $$
Step 2: Expand the numerator.
$$ (\sqrt{2} + 2\sqrt{5})(\sqrt{5} + \sqrt{2}) = \sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{2} + 2\sqrt{5}\sqrt{5} + 2\sqrt{5}\sqrt{2} $$
$$ = \sqrt{10} + 2 + 2(5) + 2\sqrt{10} $$
$$ = \sqrt{10} + 2 + 10 + 2\sqrt{10} $$
$$ = 12 + 3\sqrt{10} $$
Step 3: Expand the denominator.
Use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$:
$$ (\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2 $$
$$ = 5 - 2 $$
$$ = 3 $$
Step 4: Combine the simplified numerator and denominator.
$$ \frac{12 + 3\sqrt{10}}{3} $$
Step 5: Simplify the expression by dividing by the common factor 3.
$$ \frac{12}{3} + \frac{3\sqrt{10}}{3} $$
$$ = 4 + \sqrt{10} $$
The simplified expression is $\boxed{4 + \sqrt{10}}$.
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