To simplify the expression $\frac{\sqrt{2}}{\sqrt{5} - 1}$, we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of $\sqrt{5} - 1$ is $\sqrt{5} + 1$.

Here's the process:

$\frac{\sqrt{2}}{\sqrt{5} - 1} \times \frac{\sqrt{5} + 1}{\sqrt{5} + 1} = \frac{\sqrt{2}(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)}$

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

$(\sqrt{5} - 1)(\sqrt{5} + 1) = (\sqrt{5})^2 - (1)^2 = 5 - 1 = 4$

So, the expression becomes:

$\frac{\sqrt{2}(\sqrt{5} + 1)}{4}$

Distribute $\sqrt{2}$ in the numerator:

$\frac{\sqrt{2}\sqrt{5} + \sqrt{2}}{4}$

Which can be written as:

$\frac{\sqrt{10} + \sqrt{2}}{4}$

Thus, the simplified form of the expression $\frac{\sqrt{2}}{\sqrt{5} - 1}$ is:

$\frac{\sqrt{10} + \sqrt{2}}{4}$

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

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

Tip: Rationalizing the denominator helps eliminate radicals from the denominator, making the expression easier to work with in subsequent calculations.