\(G=\dfrac{\left(5+\sqrt{5}\right)\left(\sqrt{5}-2\right)}{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}+\dfrac{5\left(\sqrt{5}+1\right)}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}-\dfrac{3\sqrt{5}\left(3-\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}\)
\(=\dfrac{3\sqrt{5}-5}{5-4}+\dfrac{5\sqrt{5}+5}{5-1}-\dfrac{9\sqrt{5}-15}{9-5}\)
\(=3\sqrt{5}-3+\dfrac{5\sqrt{5}+5}{4}-\dfrac{9\sqrt{5}-15}{4}\)
\(=3\sqrt{5}-3+\dfrac{5\sqrt{5}+5-9\sqrt{5}+15}{4}\)
\(=3\sqrt{5}-3+\dfrac{20-4\sqrt{5}}{4}=3\sqrt{5}-3+5-\sqrt{5}\)
\(=2+2\sqrt{5}\)
\(H=\dfrac{21}{2}\left(\sqrt{4+2\sqrt{3}}+\sqrt{6-2\sqrt{5}}\right)^2-3\left(\sqrt{4-2\sqrt{3}}+\sqrt{6+2\sqrt{5}}\right)^2-15\sqrt{15}\)
\(=\dfrac{21}{2}\left(\sqrt{\left(\sqrt{3}+1\right)^2}+\sqrt{\left(\sqrt{5}-1\right)^2}\right)^2-3\left(\sqrt{\left(\sqrt{3}-1\right)^2}+\sqrt{\left(\sqrt{5}+1\right)^2}\right)^2-15\sqrt{15}\)
\(=\dfrac{21}{2}\left(\sqrt{3}+1+\sqrt{5}-1\right)^2-3\left(\sqrt{3}-1+\sqrt{5}+1\right)^2-15\sqrt{15}\)
\(=\dfrac{21}{2}\left(\sqrt{3}+\sqrt{5}\right)^2-3\left(\sqrt{3}+\sqrt{5}\right)^2-15\sqrt{15}\)
\(=\dfrac{15}{2}\left(\sqrt{3}+\sqrt{5}\right)^2-15\sqrt{15}\)
\(=\dfrac{15}{2}\left(8+2\sqrt{15}\right)-15\sqrt{15}\)
\(=60+15\sqrt{15}-15\sqrt{15}\)
\(=60\)
