\(\sqrt{4-\sqrt{9+4\sqrt{2}}}=\sqrt{4-\sqrt{\left(2\sqrt{2}\right)^2+4\sqrt{2}+1}}\)
\(=\sqrt{4-\sqrt{\left(2\sqrt{2}+1\right)^2}}\) \(=\sqrt{4-\left|2\sqrt{2}+1\right|}=\sqrt{4-2\sqrt{2}-1}\)
\(=\sqrt{3-2\sqrt{2}}=\sqrt{2-2\sqrt{2}+1}\)
\(\sqrt{\left(\sqrt{2}-1\right)^2}=\sqrt{\left|\sqrt{2}-1\right|}=\sqrt{2}-1\)