Start by trying to simplify $\sqrt{6-4\sqrt{2}}$. Let's assume there is some number $p+q\sqrt{2}$ for which
$$\sqrt{6-4\sqrt{2}} = p + q\sqrt{2}$$
Squaring both sides gives
$$6-4\sqrt{2} = (p + q\sqrt{2})^2 = p^2+2q^2 + 2pq\sqrt{2}$$
Comparing coefficients gives $6=p^2+2q^2$ and $-4=2pq$, i.e. $-2=pq$.
We need to solve $p^2+2q^2=6$ and $pq = -2$ simultaneously.
If $pq=-2$ then $q=-\frac{2}{p}$ and we can substitute this into $p^2+2q^2=6$. We get
\begin{eqnarray*}
p^2+2q^2 &=& 6 \\ \\
p^2 + 2\left(-\frac{2}{p}\right)^2 &=& 6 \\ \\
p^2 + \frac{8}{p^2} &=& 6 \\ \\
p^4+8 &=& 6p^2 \\ \\
p^4-6p^2+8 &=& 0 \\ \\
(p^2-2)(p^2-4) &=& 0
\end{eqnarray*}
Either $p^2-2=0$ or $p^2-4=0$, i.e. $p=\pm\sqrt{2}$ or $p=\pm 2$. The only valid solutions are $p = \pm2$ because we usually assume that $p$ and $q$ are rational numbers, i.e. fractions.
If $p=\pm 2$ then $pq=2$ gives $\pm 2q=-2$, and so $q=\mp 1$. Hence
$$p+q\sqrt{2} = \pm(2-\sqrt{2})$$
Recall that $\sqrt{6-4\sqrt{2}} = p + q\sqrt{2}$ and since, by definition, $\sqrt{6-4\sqrt{2}} \ge 0$ we conclude
$$\sqrt{6-4\sqrt{2}} = 2-\sqrt{2}$$
Finally:
$$\sqrt{6-4\sqrt{2}} \ \ {\color{red}{+\sqrt{2}}}= 2-\sqrt{2} \ \ {\color{red}{+\sqrt{2}}} = 2$$