\(a=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}>0\)
\(\Rightarrow a^2=4+\sqrt{10+2\sqrt{5}}-2\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}+4-\sqrt{10+2\sqrt{5}}\)
\(\Rightarrow a^2=8+2\sqrt{16-\left(10+2\sqrt{5}\right)}\)
\(\Rightarrow a^2=8+2\sqrt{6-2\sqrt{5}}\)
\(\Rightarrow a^2=8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(\Rightarrow a^2=8+2\left|\sqrt{5}-1\right|\)
\(\Rightarrow a^2=6+2\sqrt{5}\)
\(\Rightarrow a=\sqrt{6+2\sqrt{5}}=\sqrt{\left(\sqrt{5}+1\right)^2}=\left|\sqrt{5}+1\right|=1+\sqrt{5}\)
\(A=a^2-2a+12=6+2\sqrt{5}-2\left(1+\sqrt{5}\right)+12=6+2\sqrt{5}-2-2\sqrt{5}+12=16\)
