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The expression that needs to be simplified is:

\[
\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} + \frac{2\sqrt{3}}{\sqrt{6} + 2}
\]

Let's break this down step by step.

### Step 1: Rationalize the Denominators
The first two fractions involve square roots in the denominator, so we will rationalize each of them.

- For the first term, \(\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}}\), multiply the numerator and denominator by \(\sqrt{6} + \sqrt{3}\).
  
- For the second term, \(\frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}}\), multiply the numerator and denominator by \(\sqrt{6} + \sqrt{2}\).

The third term already has a rational denominator, so we leave that as is.

Let me compute these steps for you.After rationalizing the denominators, the simplified terms are:

1. \(\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} = \sqrt{6} + 2\sqrt{3}\)
2. \(\frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} = \sqrt{6} + 3\sqrt{2}\)
3. The third term remains: \(\frac{2\sqrt{3}}{\sqrt{6} + 2} = \frac{2\sqrt{3}}{2 + \sqrt{6}}\)

Now, we combine these three simplified terms:

\[
(\sqrt{6} + 2\sqrt{3}) - (\sqrt{6} + 3\sqrt{2}) + \frac{2\sqrt{3}}{2 + \sqrt{6}}
\]

I will proceed to simplify the expression further.The entire expression simplifies to **0**.

This means the given expression:

\[
\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} + \frac{2\sqrt{3}}{\sqrt{6} + 2}
\]

is equivalent to zero after simplification.

Would you like further details or have any other questions? Here are a few related questions for practice:

1. How do you rationalize the denominator of a fraction?
2. What is the process to simplify square root expressions?
3. Why is multiplying by the conjugate useful in rationalization?
4. Can you simplify \(\frac{5}{\sqrt{2} + 1}\) using a similar method?
5. How do different square root expressions behave when combined?

**Tip:** Rationalizing denominators often involves multiplying by the conjugate to eliminate square roots from the denominator.

Simplify: \(\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}} + \frac{2\sqrt{3}}{\sqrt{6} + 2}\)

Learn how to simplify a complex expression involving radicals and fractions. The problem demonstrates rationalization of square roots and simplifying terms. Suitable for high school students (Grades 10-12).

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