\(n=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}\)
\(\Rightarrow n^2=\left(3+\sqrt{5+2\sqrt{3}}\right)+2\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}+\left(3-\sqrt{5+2\sqrt{3}}\right)\)
\(=6+2\sqrt{9-\left(5+2\sqrt{3}\right)}\)
\(=6+2\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=6+2\sqrt{3}-2=\left(\sqrt{3}+1\right)^2\)
\(\Rightarrow n=\sqrt{3}+1\)
\(\Rightarrow n^2-2n-2=\left(4+2\sqrt{3}\right)-2\left(\sqrt{3}+1\right)-2=0\)
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