\(A=\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{2+\sqrt{6+2\sqrt{5}}}+\frac{\sqrt{2}\left(1-\sqrt{5}\right)}{2-\sqrt{6-2\sqrt{5}}}=\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{2+\sqrt{\left(\sqrt{5}+1\right)^2}}+\frac{\sqrt{2}\left(1-\sqrt{5}\right)}{2-\sqrt{\left(\sqrt{5}-1\right)^2}}\)
\(=\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{2+\sqrt{5}+1}+\frac{\sqrt{2}\left(1-\sqrt{5}\right)}{2-\sqrt{5}+1}=\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{3+\sqrt{5}}+\frac{\sqrt{2}\left(1-\sqrt{5}\right)}{3-\sqrt{5}}\)
\(=\sqrt{2}\left(\frac{\left(1+\sqrt{5}\right)\left(3-\sqrt{5}\right)+\left(1-\sqrt{5}\right)\left(3+\sqrt{5}\right)}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}\right)=\sqrt{2}\left(\frac{-2+2\sqrt{5}-2-2\sqrt{5}}{4}\right)=-\sqrt{2}\)
