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.