$\begin{array}{l} x \in \left( {0;\pi } \right) \Rightarrow \dfrac{x}{4} \in \left( {0;\dfrac{\pi }{4}} \right)\\ \Rightarrow 0 \le \sin \left( {\dfrac{x}{4}} \right) \le \dfrac{{\sqrt 2 }}{2} \Rightarrow 0 \le {\sin ^2}\left( {\dfrac{x}{4}} \right) \le \dfrac{1}{2}\\ \left[ {{{\sin }^4}\left( {\dfrac{x}{8}} \right) + {{\cos }^4}\left( {\dfrac{x}{8}} \right)} \right] = 2m - 1\\ \Leftrightarrow {\left[ {{{\sin }^2}\left( {\dfrac{x}{8}} \right) + {{\cos }^2}\left( {\dfrac{x}{8}} \right)} \right]^2} - 2{\sin ^2}\left( {\dfrac{x}{8}} \right){\cos ^2}\left( {\dfrac{x}{8}} \right) = 2m - 1\\ \Leftrightarrow 1 - \dfrac{1}{2}{\sin ^2}\left( {\dfrac{x}{4}} \right) = 2m - 1\\ \Leftrightarrow \dfrac{1}{2}{\sin ^2}\left( {\dfrac{x}{4}} \right) = 2 - 2m\\ \Leftrightarrow {\sin ^2}\left( {\dfrac{x}{4}} \right) = 4 - 4m\\ \text{Phương trình có nghiệm} \Leftrightarrow 0 \le 4 - 4m \le \dfrac{1}{2}\\ \Leftrightarrow \dfrac{7}{8} \le m \le 1 \end{array}$
