\(\dfrac{\sqrt{2}\left(3+\sqrt{5}\right)}{\sqrt{2}\left(\sqrt{10}+\sqrt{3+\sqrt{5}}\right)}-\dfrac{\sqrt{2}\left(3-\sqrt{5}\right)}{\sqrt{2}\left(\sqrt{10}+\sqrt{3-\sqrt{5}}\right)}\)
=\(\dfrac{3\sqrt{2}+\sqrt{10}}{\sqrt{20}+\sqrt{6+2\sqrt{5}}}-\dfrac{3\sqrt{2}-\sqrt{10}}{\sqrt{20}+\sqrt{6-2\sqrt{5}}}\)
=\(\dfrac{3\sqrt{2}+\sqrt{10}}{\sqrt{20}+\sqrt{5+2\sqrt{5}+1}}-\dfrac{3\sqrt{2}-\sqrt{10}}{\sqrt{20}+\sqrt{5-2\sqrt{5}+1}}\)
=\(\dfrac{3\sqrt{2}+\sqrt{10}}{\sqrt{20}+\sqrt{\left(\sqrt{5}+1\right)^2}}-\dfrac{3\sqrt{2}-\sqrt{10}}{\sqrt{20}+\sqrt{\left(\sqrt{5}-1\right)^2}}\)
=\(\dfrac{3\sqrt{2}+\sqrt{10}}{\sqrt{20}+\sqrt{5}+1}-\dfrac{3\sqrt{2}-\sqrt{10}}{\sqrt{20}+\sqrt{5}-1}\)
=\(\dfrac{3\sqrt{2}+\sqrt{10}}{3\sqrt{5}+1}-\dfrac{3\sqrt{2}-\sqrt{10}}{3\sqrt{5}-1}\)
=\(\dfrac{(3\sqrt{2}+\sqrt{10})\left(3\sqrt{5}-1\right)}{(3\sqrt{5}+1)\left(3\sqrt{5}-1\right)}-\dfrac{(3\sqrt{2}-\sqrt{10})\left(3\sqrt{5}+1\right)}{(3\sqrt{5}+1)\left(3\sqrt{5}-1\right)}\)
=\(\dfrac{9\sqrt{10}-3\sqrt{2}+3\sqrt{50}-\sqrt{10}-\left(9\sqrt{10}+3\sqrt{2}-3\sqrt{50}-\sqrt{10}\right)}{(3\sqrt{5}+1)\left(3\sqrt{5}-1\right)}\)
=\(\dfrac{-6\sqrt{2}+6\sqrt{50}}{\left(3\sqrt{5}\right)^2-1}\)=\(\dfrac{-6\left(\sqrt{2}-\sqrt{50}\right)}{45-1}\)
=\(\dfrac{-6\sqrt{2}\left(1-5\right)}{44}=\dfrac{24\sqrt{2}}{44}=\dfrac{6\sqrt{2}}{11}\)