To simplify the expression $\left( \frac{\sqrt{5} + \sqrt{2}}{2\sqrt{5} - \sqrt{2}} \right)^2$, we can proceed in steps as follows:

1. Rewrite the expression: $\left( \frac{\sqrt{5} + \sqrt{2}}{2\sqrt{5} - \sqrt{2}} \right)^2$

2. Simplify the fraction inside the square by rationalizing the denominator. To do this, multiply both the numerator and the denominator by the conjugate of the denominator, $2\sqrt{5} + \sqrt{2}$:

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

3. Expand both the numerator and the denominator:

   • In the numerator: $(\sqrt{5} + \sqrt{2})(2\sqrt{5} + \sqrt{2}) = \sqrt{5} \cdot 2\sqrt{5} + \sqrt{5} \cdot \sqrt{2} + \sqrt{2} \cdot 2\sqrt{5} + \sqrt{2} \cdot \sqrt{2}$ Simplifying each term: [ = 2 \cdot 5 + \sqrt{10} + 2\sqrt{10} + 2 = 10