The expression you provided is:

\[
\frac{3\sqrt{2}}{4 + 2\sqrt{5}}
\]

To simplify this, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of \(4 + 2\sqrt{5}\) is \(4 - 2\sqrt{5}\).

1. Multiply both the numerator and the denominator by \(4 - 2\sqrt{5}\):

\[
\frac{3\sqrt{2}}{4 + 2\sqrt{5}} \times \frac{4 - 2\sqrt{5}}{4 - 2\sqrt{5}} = \frac{3\sqrt{2}(4 - 2\sqrt{5})}{(4 + 2\sqrt{5})(4 - 2\sqrt{5})}
\]

2. First, simplify the denominator using the difference of squares formula \((a + b)(a - b) = a^2 - b^2\):

\[
(4 + 2\sqrt{5})(4 - 2\sqrt{5}) = 4^2 - (2\sqrt{5})^2 = 16 - 20 = -4
\]

3. Now, simplify the numerator:

\[
3\sqrt{2}(4 - 2\sqrt{5}) = 12\sqrt{2} - 6\sqrt{10}
\]

4. Putting it all together, we now have:

\[
\frac{12\sqrt{2} - 6\sqrt{10}}{-4} = -3\sqrt{2} + \frac{3\sqrt{10}}{2}
\]

So, the simplified expression is:

\[
-3\sqrt{2} + \frac{3\sqrt{10}}{2}
\]

Would you like further details or have any questions?

Here are five related questions you might find interesting:
1. How do you rationalize other complex denominators?
2. What is the difference between a rational and an irrational number?
3. Can you apply the difference of squares to any radical expression?
4. What is the general form of a conjugate and how is it useful in simplifications?
5. How would this simplification change if there were additional radicals in the numerator?

**Tip:** Always multiply by the conjugate to eliminate square roots from the denominator when rationalizing!

Learn how to simplify the radical expression 3√2 / (4 + 2√5) by rationalizing the denominator and using the difference of squares formula. Follow a step-by-step explanation, suitable for students in Grades 10-12, to understand radical expressions and conjugate multiplication.

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