$\begin{align}& \,\,\,\,\,2\sqrt{4+\sqrt{6-2\sqrt{5}}}\left( \sqrt{10}-\sqrt{2} \right) \\ & =2\sqrt{4+\sqrt{5-2\sqrt{5}+1}}\left( \sqrt{10}-\sqrt{2} \right) \\ & =2\sqrt{4+\sqrt{{{\left( \sqrt{5}-1 \right)}^{2}}}}\left( \sqrt{10}-\sqrt{2} \right) \\ & =2\sqrt{4+\sqrt{5}-1}\left( \sqrt{10}-\sqrt{2} \right) \\ & =2\sqrt{3+\sqrt{5}}\left( \sqrt{10}-\sqrt{2} \right) \\ & =\sqrt{2}.\sqrt{2\left( 3+\sqrt{5} \right)}\left( \sqrt{10}-\sqrt{2} \right) \\ & =\sqrt{2}.\sqrt{6+2\sqrt{5}}\left( \sqrt{10}-\sqrt{2} \right) \\ & =\sqrt{2}.\left( \sqrt{5}+1 \right)\left( \sqrt{10}-\sqrt{2} \right) \\ & =\left( \sqrt{10}+\sqrt{2} \right)\left( \sqrt{10}-\sqrt{2} \right) \\ & =10-2 \\ & =8 \\ \end{align}$