[tex]\displaystyle\\
\sqrt{6-4\sqrt2}+\sqrt{6+4\sqrt2} = \\\\
=\sqrt{4+2-2\times 2\times\sqrt2}+\sqrt{4+2+2\times 2\times\sqrt2} = \\\\
=\sqrt{4-2\times 2\times\sqrt2 +2}+\sqrt{4+2\times 2\times\sqrt2+2} = \\\\
=\sqrt{2^2-2\times 2\times\sqrt2 +(-\sqrt2)^2}+\sqrt{2^2+2\times 2\times\sqrt2+(\sqrt2)^2} = \\\\
=\sqrt{(2-\sqrt2)^2}+\sqrt{(2+\sqrt2)^2} = \\\\
= 2-\underline{\sqrt2} + 2+\underline{\sqrt2} = 2 + 2 = \boxed{4 \in \mathbb{Q}}
[/tex]