\(\sqrt{7-4\sqrt{3}}+\sqrt{2\sqrt{3}+4}-\sqrt{6\sqrt{3}+12}=\sqrt{3-4\sqrt{3}+4}+\sqrt{1+2\sqrt{3}+3}-\sqrt{9+6\sqrt{3}+3}\)
\(=\sqrt{\left(2-\sqrt{3}\right)^2}+\sqrt{\left(1+\sqrt{3}\right)^2}-\sqrt{\left(3+\sqrt{3}\right)^2}=\left|2-\sqrt{3}\right|+\left|1+\sqrt{3}\right|-\left|3+\sqrt{3}\right|\)
\(=2-\sqrt{3}+1+\sqrt{3}-3-\sqrt{3}=-\sqrt{3}\)
\(=\sqrt{4-2.2\sqrt{3}+3}+\sqrt{3+2\sqrt{3}+1}-\sqrt{9+2.3.\sqrt{3}+3}\)
\(=\sqrt{\left(2-\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{3}+1\right)^2}+\sqrt{\left(3+\sqrt{3}\right)^2}\)
= \(2-\sqrt{3}+\sqrt{3}+1+3+\sqrt{3}=7+4\sqrt{3}\)
