\(A=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
\(A^2=4+\sqrt{10+2\sqrt{5}}+2.\sqrt{4+\sqrt{10+2\sqrt{5}}}.\sqrt{4-\sqrt{10+2\sqrt{5}}}+4-\sqrt{10+2\sqrt{5}}\)
\(A^2=8+2.\sqrt{16-10-2\sqrt{5}}\)
\(A^2=8+2.\sqrt{6-2\sqrt{5}}\)
\(A^2=8+2.\sqrt{5-2\sqrt{5}+1}\)
\(A^2=8+2.\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(A^2=8+2.\left(\sqrt{5}-1\right)\)
\(A^2=8+2\sqrt{5}-2=6+2\sqrt{5}\)
\(A^2=5+2\sqrt{5}+1=\left(\sqrt{5}+1\right)^2\)
\(\Rightarrow A=\sqrt{5}+1\)
:))
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