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