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

The conjugate of $\sqrt{2} - \sqrt{5}$ is $\sqrt{2} + \sqrt{5}$.

Here's the process:

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

Now, simplify the denominator using the difference of squares formula $(a - b)(a + b) = a^2 - b^2$:

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

So, the expression becomes:

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

Distribute $\sqrt{3}$ in the numerator:

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

This can be written as:

$\frac{\sqrt{6} + \sqrt{15}}{-3}$

Finally, we can write this as:

$-\frac{\sqrt{6} + \sqrt{15}}{3}$

Thus, the simplified form of the expression $\frac{\sqrt{3}}{\sqrt{2} - \sqrt{5}}$ is:

$-\frac{\sqrt{6} + \sqrt{15}}{3}$

Would you like more details or have any questions? Here are some related problems:

1. Simplify $\frac{\sqrt{7}}{\sqrt{3} - \sqrt{2}}$.
2. Rationalize the denominator of $\frac{\sqrt{5}}{\sqrt{3} + \sqrt{7}}$.
3. Simplify $\frac{\sqrt{8}}{\sqrt{6} - \sqrt{2}}$.
4. Rationalize $\frac{1}{\sqrt{2} + \sqrt{5}}$.
5. Simplify $\frac{\sqrt{10}}{\sqrt{5} - \sqrt{3}}$.

Tip: Always check your simplified expression by multiplying it back out to ensure it matches the original expression.