[tex]\displaystyle\\ \sqrt{6+4\sqrt{2}}+ \sqrt{6-4\sqrt{2}}=\\\\= \sqrt{6+2\times2\times\sqrt{2}}+\sqrt{6-2\times2\times\sqrt{2}}=\\\\= \sqrt{4+2+2\times2\times\sqrt{2}}+\sqrt{4+2-2\times2\times\sqrt{2}}=\\\\= \sqrt{4+2\times2\times\sqrt{2}+2}+\sqrt{4-2\times2\times\sqrt{2}+2}=\\\\= \sqrt{2^2+2\times2\times\sqrt{2}+\Big(\sqrt{2}\Big)^2}+\sqrt{2^2-2\times2\times\sqrt{2}+\Big(\sqrt{2}\Big)^2}=[/tex]
[tex]\displaystyle\\= \sqrt{\Big(2+\sqrt{2}\Big)^2 }+ \sqrt{\Big(2-\sqrt{2}\Big)^2}=\\\\=\Big(2+\sqrt{2}\Big)+\Big(2-\sqrt{2}\Big)=\\\\2+\underline{\sqrt{2}}+2-\underline{\sqrt{2}}=2+2=\boxed{\bf4\in Q}[/tex]
