\(a=\sqrt{2}+\sqrt{7-\sqrt{21+4\sqrt{5}}}+1\)
\(=\sqrt{2}+\sqrt{7-\sqrt{1^2+\left(\sqrt{20}\right)^2+2.\sqrt{20}.1}}+1\)
\(=\sqrt{2}+\sqrt{7-\sqrt{\left(\sqrt{20}+1\right)^2}}+1=\sqrt{2}+\sqrt{6-2\sqrt{5}}+1\)\(=\sqrt{2}+\sqrt{1^2+\left(\sqrt{5}\right)^2-2.1.\sqrt{5}}+1\)
\(=\sqrt{2}+\sqrt{\left(\sqrt{5}-1\right)^2}+1=\sqrt{5}+\sqrt{2}\)
\(\Leftrightarrow a^2=7+2\sqrt{10}\)
\(\Leftrightarrow\left(a^2-7\right)^2=40\)
\(\Leftrightarrow a^4-14a^2+49-40=0\Leftrightarrow a^4-14a^2+9=0\)
\(\Leftrightarrow\left(a^4-14a^2+10\right)-1=0\Leftrightarrow a^4-14a^2+10=1\)
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