Let's simplify the given expression:

$\left(\frac{\sqrt{5} + \sqrt{2}}{2\sqrt{5} - \sqrt{2}}\right)^2$

Step 1: Rationalize the denominator

To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, $2\sqrt{5} + \sqrt{2}$.

$\left(\frac{\sqrt{5} + \sqrt{2}}{2\sqrt{5} - \sqrt{2}}\right) \times \frac{2\sqrt{5} + \sqrt{2}}{2\sqrt{5} + \sqrt{2}} = \frac{(\sqrt{5} + \sqrt{2})(2\sqrt{5} + \sqrt{2})}{(2\sqrt{5} - \sqrt{2})(2\sqrt{5} + \sqrt{2})}$

Step 2: Simplify the denominator

The denominator simplifies as a difference of squares:

$(2\sqrt{5})^2 - (\sqrt{2})^2 = 4 \times 5 - 2 = 20 - 2 = 18$

So, the denominator is 18.

Step 3: Simplify the numerator

Now, expand the numerator:

$(\sqrt{5} + \sqrt{2})(2\sqrt{5} + \sqrt{2}) = \sqrt{5}(2\sqrt{5}) + \sqrt{5}(\sqrt{2}) + \sqrt{2}(2\sqrt{5}) + \sqrt{2}(\sqrt{2})$

Simplifying each term:

$\sqrt{5}(2\sqrt{5}) = 2 \times 5 = 10$ $\sqrt{5}(\sqrt{2}) = \sqrt{10}$ $\sqrt{2}(2\sqrt{5}) = 2\sqrt{10}$ $\sqrt{2}(\sqrt{2}) = 2$

So, the numerator becomes:

$10 + \sqrt{10} + 2\sqrt{10} + 2 = 12 + 3\sqrt{10}$

Step 4: Form the simplified expression

Now, the expression is:

$\frac{12 + 3\sqrt{10}}{18}$

Step 5: Further simplification

We can divide both terms in the numerator by 18:

$\frac{12}{18} + \frac{3\sqrt{10}}{18} = \frac{2}{3} + \frac{\sqrt{10}}{6}$

Step 6: Square the expression

Now, we need to square this result:

$\left(\frac{2}{3} + \frac{\sqrt{10}}{6}\right)^2 = \left(\frac{2}{3}\right)^2 + 2\left(\frac{2}{3}\right)\left(\frac{\sqrt{10}}{6}\right) + \left(\frac{\sqrt{10}}{6}\right)^2$

Simplify each term:

$\left(\frac{2}{3}\right)^2 = \frac{4}{9}$ $2\left(\frac{2}{3}\right)\left(\frac{\sqrt{10}}{6}\right) = \frac{4\sqrt{10}}{18} = \frac{2\sqrt{10}}{9}$ $\left(\frac{\sqrt{10}}{6}\right)^2 = \frac{10}{36} = \frac{5}{18}$

Step 7: Combine all terms

Now, sum the three terms:

$\frac{4}{9} + \frac{2\sqrt{10}}{9} + \frac{5}{18} = \frac{8}{18} + \frac{2\sqrt{10}}{9} + \frac{5}{18}$

$= \frac{13}{18} + \frac{2\sqrt{10}}{9}$

Thus, the final simplified result is:

$\frac{13}{18} + \frac{2\sqrt{10}}{9}$

Would you like more details or have any questions about this solution?

Here are 5 related questions to deepen understanding:

1. How do you rationalize the denominator of a radical expression?
2. What is the difference of squares formula and how is it applied?
3. Can you explain how to expand binomial products involving square roots?
4. How does squaring a binomial work in general?
5. What other methods can simplify complex radical expressions?

Tip: Rationalizing the denominator is crucial in simplifying expressions that involve square roots, and it often involves multiplying by the conjugate.