\(\frac{\sqrt{\left(3\sqrt{3}-4\right)\left(2\sqrt{3}+1\right)}}{2\sqrt{3}+1}-\frac{\sqrt{\left(\sqrt{3}+4\right)\left(5-2\sqrt{3}\right)}}{5-2\sqrt{3}}=\frac{\sqrt{14-5\sqrt{3}}}{2\sqrt{3}+1}-\frac{\sqrt{14-3\sqrt{3}}}{5-2\sqrt{3}}\)
\(=\frac{\sqrt{28-10\sqrt{3}}}{\sqrt{2}\left(2\sqrt{3}+1\right)}-\frac{\sqrt{28-6\sqrt{3}}}{\sqrt{2}\left(5-2\sqrt{3}\right)}=\frac{5-\sqrt{3}}{\sqrt{2}\left(2\sqrt{3}+1\right)}-\frac{3\sqrt{3}-1}{\sqrt{2}\left(5-2\sqrt{3}\right)}\)
\(=\frac{1}{\sqrt{2}}\left(\frac{\left(5-\sqrt{3}\right)\left(2\sqrt{3}-1\right)}{11}-\frac{\left(3\sqrt{3}-1\right)\left(5+2\sqrt{3}\right)}{13}\right)\)
\(=\frac{1}{\sqrt{2}}\left(\frac{11\sqrt{3}-11}{11}-\frac{13\sqrt{3}+13}{13}\right)=\frac{1}{\sqrt{2}}\left(\sqrt{3}-1-\sqrt{3}-1\right)=-\sqrt{2}\)
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