$$\eqalign{
& y = \sqrt {5\sin 4x - 6\cos 4x + 2m - 1} \cr
& DKXD:\,\,\,5\sin 4x - 6\cos 4x + 2m - 1 \ge 0\,\,\forall x \cr
& \Rightarrow {5 \over {\sqrt {61} }}\sin 4x - {6 \over {\sqrt {41} }}\cos 4x \ge {{1 - 2m} \over {\sqrt {61} }} \cr
& Dat\,\,\left\{ \matrix{
{5 \over {\sqrt {61} }} = \cos \alpha \hfill \cr
{6 \over {\sqrt {61} }} = \sin \alpha \hfill \cr} \right. \cr
& \Rightarrow \sin 4x\cos \alpha - \cos 4x\sin \alpha \ge {{1 - 2m} \over {\sqrt {61} }}\,\,\forall x \cr
& \Leftrightarrow \sin \left( {4x - \alpha } \right) \ge {{1 - 2m} \over {\sqrt {61} }}\,\,\forall x \cr
& \Leftrightarrow - 1 \ge {{1 - 2m} \over {\sqrt {61} }} \cr
& \Leftrightarrow - \sqrt {61} \ge 1 - 2m \cr
& \Leftrightarrow - \sqrt {61} - 1 \ge - 2m \cr
& \Leftrightarrow \sqrt {61} + 1 \le m \cr} $$
