Ta có: \(\left(\frac{3}{\sqrt{5}-\sqrt{2}}+\frac{4}{\sqrt{6}+\sqrt{2}}\right)\cdot\left(\sqrt{3}-1\right)^2\)
\(=\left(\frac{3\cdot\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)}+\frac{4\left(\sqrt{5}-\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}\right)\cdot\left(\sqrt{3}-1\right)^2\)
\(=\frac{3\sqrt{6}+3\sqrt{2}+4\sqrt{5}-4\sqrt{2}}{\left(\sqrt{5}-\sqrt{2}\right)\cdot\sqrt{2}\cdot\left(\sqrt{3}+1\right)}\cdot\left(\sqrt{3}-1\right)^2\)
\(=\frac{3\sqrt{6}-\sqrt{2}+4\sqrt{5}}{\left(2\sqrt{5}-2\right)\left(\sqrt{3}+1\right)}\cdot\left(4-2\sqrt{3}\right)\)
\(=\frac{\sqrt{2}\left(3\sqrt{3}-1+2\sqrt{10}\right)\cdot\left(4-2\sqrt{3}\right)}{\sqrt{2}\cdot\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{3}+1\right)}\)
\(=\frac{12\sqrt{3}-6-4+2\sqrt{3}+8\sqrt{10}-4\sqrt{30}}{\sqrt{15}+\sqrt{5}-\sqrt{6}-\sqrt{2}}\)
\(=\frac{14\sqrt{3}-10+8\sqrt{10}-4\sqrt{30}}{\sqrt{15}+\sqrt{5}-\sqrt{6}-\sqrt{2}}\)
\(=\left(\frac{3\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}+\frac{4\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}\right)\left(\sqrt{3}-1\right)^2\)
\(=\left(\frac{3\left(\sqrt{5}+\sqrt{2}\right)}{3}+\frac{4\left(\sqrt{6}-\sqrt{2}\right)}{4}\right)\left(\sqrt{3}-1\right)^2\)
\(=\left(\sqrt{5}+\sqrt{2}+\sqrt{6}-\sqrt{2}\right)\left(\sqrt{3}-1\right)^2\)
\(=\left(\sqrt{6}+\sqrt{5}\right)\left(\sqrt{3}-1\right)^2\)
\(\left(=4\sqrt{5}+4\sqrt{6}-6\sqrt{2}-2\sqrt{15}\right)\)
