$A=(4+\sqrt{15})(\sqrt{10}-\sqrt{6})\sqrt{4-\sqrt{15}}
\\\Rightarrow 2A=(8+2\sqrt{15})(\sqrt{5}-\sqrt{3})\sqrt{8-2\sqrt{15}}
\\=(5+2\sqrt{15}+3)(\sqrt{5}-\sqrt{3})\sqrt{5-2\sqrt{15}+3}
\\=(\sqrt{5}+\sqrt{3})^2(\sqrt{5}-\sqrt{3})\sqrt{(\sqrt{5}-\sqrt{3})^2}
\\=(\sqrt{5}+\sqrt{3})^2(\sqrt{5}-\sqrt{3})^2=[(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})]^2
\\=(5-3)^2=2^2=4
\\\Rightarrow A=2$