Simplify the expression $\sqrt{-6-2\sqrt{5}}$.

1. Problem Description

Simplify the expression $\sqrt{-6-2\sqrt{5}}$.

2. Solution Steps

First, we observe that the expression inside the square root is negative. We can rewrite the expression as:

$\sqrt{-6-2\sqrt{5}} = \sqrt{-(6+2\sqrt{5})} = \sqrt{-1} \cdot \sqrt{6+2\sqrt{5}} = i \cdot \sqrt{6+2\sqrt{5}}$

Now, we want to simplify $\sqrt{6+2\sqrt{5}}$. We look for two numbers $a$ and $b$ such that $a+b=6$ and $ab=5$. We see that $a=5$ and $b=1$ satisfy these conditions. Therefore, we can write $6+2\sqrt{5}$ as $(\sqrt{5} + 1)^2$.

$6 + 2\sqrt{5} = 5 + 1 + 2\sqrt{5 \cdot 1} = (\sqrt{5})^2 + 1^2 + 2 \cdot \sqrt{5} \cdot 1 = (\sqrt{5}+1)^2$

So, $\sqrt{6+2\sqrt{5}} = \sqrt{(\sqrt{5}+1)^2} = \sqrt{5}+1$.

Then, $\sqrt{-6-2\sqrt{5}} = i \sqrt{6+2\sqrt{5}} = i (\sqrt{5}+1) = i(\sqrt{5}+1)$.

Alternatively, one may have assumed there's a typo and the problem meant $\sqrt{6-2\sqrt{5}}$. In that case:

$6-2\sqrt{5} = 5+1-2\sqrt{5} = (\sqrt{5})^2 + 1^2 - 2\sqrt{5} = (\sqrt{5}-1)^2$

$\sqrt{6-2\sqrt{5}} = \sqrt{(\sqrt{5}-1)^2} = \sqrt{5}-1$.

Based on the problem presented in the image, however, the presence of a negative sign makes $\sqrt{-6-2\sqrt{5}}$ a complex number.

3. Final Answer

$i(\sqrt{5}+1)$