The expression $\frac{3\sqrt{2}}{\sqrt{5} - 2\sqrt{2}}$ can be simplified by rationalizing the denominator. Here's how:

Step 1: Multiply by the conjugate

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{5} - 2\sqrt{2}$ is $\sqrt{5} + 2\sqrt{2}$.

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

Step 2: Simplify the numerator

Distribute $3\sqrt{2}$ in the numerator:

$3\sqrt{2} \times \sqrt{5} + 3\sqrt{2} \times 2\sqrt{2} = 3\sqrt{10} + 6 \times 2 = 3\sqrt{10} + 12$

So, the numerator is $3\sqrt{10} + 12$.

Step 3: Simplify the denominator

Use the difference of squares formula $(a - b)(a + b) = a^2 - b^2$:

$(\sqrt{5})^2 - (2\sqrt{2})^2 = 5 - 8 = -3$

Step 4: Write the final expression

Now, the expression simplifies to:

$\frac{3\sqrt{10} + 12}{-3}$

This can be further simplified by dividing each term in the numerator by $-3$:

$\frac{3\sqrt{10}}{-3} + \frac{12}{-3} = -\sqrt{10} - 4$

Final Answer:

$\frac{3\sqrt{2}}{\sqrt{5} - 2\sqrt{2}} = -\sqrt{10} - 4$

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:

1. How do you rationalize the denominator in more complex expressions?
2. What is the difference of squares and when is it used?
3. How can conjugates be used in simplifying radical expressions?
4. What are the properties of square roots that help in simplifying expressions?
5. Can the expression $\sqrt{a} + \sqrt{b}$ be further simplified?

Tip: Always look for opportunities to use conjugates to simplify expressions involving square roots in the denominator.