\(\left(\sqrt{5}-2\right)\cdot\sqrt{3+\sqrt{5}}+\left(\sqrt{5}+2\right)\cdot\sqrt{3-\sqrt{5}}\)
\(=\dfrac{\left(\sqrt{5}-2\right)\cdot\sqrt{2}\cdot\sqrt{3+\sqrt{5}}+\left(\sqrt{5}+2\right)\cdot\sqrt{2}\cdot\sqrt{3-\sqrt{5}}}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{5}-2\right)\cdot\sqrt{6+2\sqrt{5}}+\left(\sqrt{5}+2\right)\cdot\sqrt{6-2\sqrt{5}}}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{5}-2\right)\sqrt{\left(\sqrt{5}+1\right)^2}+\left(\sqrt{5}+2\right)\cdot\sqrt{\left(\sqrt{5}-1\right)^2}}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{5}-2\right)\left(\sqrt{5}+1\right)+\left(\sqrt{5}+2\right)\left(\sqrt{5}-1\right)}{\sqrt{2}}\)
\(=\dfrac{5+\sqrt{5}-2\sqrt{5}-2+5-\sqrt{5}+2\sqrt{5}-2}{\sqrt{2}}\)
\(=\dfrac{6}{\sqrt{2}}=3\sqrt{2}\)