To simplify the expression $\frac{\sqrt{5} + 2}{\sqrt{3} - \sqrt{2}}$, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of $\sqrt{3} - \sqrt{2}$ is $\sqrt{3} + \sqrt{2}$.

Let's proceed with the steps:

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

Step 1: Simplify the denominator

The denominator is a difference of squares: $(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1$

Step 2: Expand the numerator

Now, expand the numerator: $(\sqrt{5} + 2)(\sqrt{3} + \sqrt{2}) = \sqrt{5} \cdot \sqrt{3} + \sqrt{5} \cdot \sqrt{2} + 2 \cdot \sqrt{3} + 2 \cdot \sqrt{2}$

This simplifies to: $\sqrt{15} + \sqrt{10} + 2\sqrt{3} + 2\sqrt{2}$

Final Expression:

Since the denominator is 1, the simplified expression is: $\sqrt{15} + \sqrt{10} + 2\sqrt{3} + 2\sqrt{2}$

This is the final simplified form of the expression.

Would you like further clarification or any details on the steps?

Here are 5 related questions to deepen your understanding:

1. How do you multiply two expressions involving square roots?
2. What is the purpose of rationalizing the denominator?
3. Can you explain the difference of squares used in rationalizing denominators?
4. How would you simplify $\frac{3 + \sqrt{7}}{2 - \sqrt{5}}$?
5. What happens if the denominator is already rationalized?

Tip: Always use the conjugate of a binomial with square roots in the denominator to eliminate irrational numbers.