To simplify the given expression:

$\frac{\sqrt{5} + 2\sqrt{3}}{\sqrt{5} - 3\sqrt{5}}$

First, simplify the denominator:

$\sqrt{5} - 3\sqrt{5} = -2\sqrt{5}$

Thus, the expression becomes:

$\frac{\sqrt{5} + 2\sqrt{3}}{-2\sqrt{5}}$

Now, separate the terms in the numerator:

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

Simplify each term:

1. $\frac{\sqrt{5}}{-2\sqrt{5}} = -\frac{1}{2}$
2. $\frac{2\sqrt{3}}{-2\sqrt{5}} = -\frac{\sqrt{3}}{\sqrt{5}}$

Thus, the simplified expression is:

$-\frac{1}{2} - \frac{\sqrt{3}}{\sqrt{5}}$

If you'd like, you can rationalize the second term by multiplying both the numerator and denominator by $\sqrt{5}$:

$-\frac{\sqrt{3}}{\sqrt{5}} = -\frac{\sqrt{15}}{5}$

So, the final simplified expression is:

$-\frac{1}{2} - \frac{\sqrt{15}}{5}$

Would you like more details or clarification on any step?

Here are 5 related questions:

1. How do you rationalize a denominator?
2. What happens if the denominator contains multiple square roots?
3. How do you simplify terms with different radicals?
4. How can this approach be extended to complex fractions?
5. How would the expression change if both the numerator and denominator had radicals?

Tip: Always simplify square roots and like terms in both the numerator and denominator before proceeding to rationalize.