To simplify \(\frac{6 - 3\sqrt{5}}{2 + \sqrt{5}}\), we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + \sqrt{5}\) is \(2 - \sqrt{5}\).

Let's go through the steps:

1. **Multiply the numerator and denominator by the conjugate of the denominator:**
   \[
   \frac{6 - 3\sqrt{5}}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{(6 - 3\sqrt{5})(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})}
   \]

2. **Simplify the denominator:**
   \[
   (2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
   \]

3. **Expand the numerator:**
   \[
   (6 - 3\sqrt{5})(2 - \sqrt{5}) = 6 \times 2 - 6 \times \sqrt{5} - 3\sqrt{5} \times 2 + 3\sqrt{5} \times \sqrt{5}
   \]
   Simplifying each term:
   \[
   12 - 6\sqrt{5} - 6\sqrt{5} + 15 = 12 - 12\sqrt{5} + 15
   \]
   Combine like terms:
   \[
   27 - 12\sqrt{5}
   \]

4. **Final simplified expression:**
   Now, we have:
   \[
   \frac{27 - 12\sqrt{5}}{-1} = -27 + 12\sqrt{5}
   \]

Hence, the simplified form of the given expression is:

\[
-27 + 12\sqrt{5}
\]

Do you want more details or have any questions? 

Here are some related problems:

1. Simplify \(\frac{4 + 2\sqrt{3}}{3 - \sqrt{3}}\).
2. Rationalize the denominator of \(\frac{5 - \sqrt{2}}{3 + \sqrt{2}}\).
3. Simplify \(\frac{7 - 3\sqrt{2}}{2 - \sqrt{2}}\).
4. Simplify \(\frac{1 + \sqrt{6}}{2 + \sqrt{3}}\).
5. Find the conjugate of \(3 + 2\sqrt{7}\).

**Tip:** The conjugate of a binomial expression \(a + b\sqrt{c}\) is \(a - b\sqrt{c}\). Multiplying by the conjugate helps to eliminate the square root in the denominator.

This detailed solution simplifies the expression (6 - 3√5) / (2 + √5) by multiplying the numerator and denominator by the conjugate of the denominator. Learn how to apply the conjugate theorem and the difference of squares, making this problem ideal for students in Grades 9-12.

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