$\begin{array}{l}
2\left( {\cos 4x - \sin 4x} \right) + \sqrt 3 = 0\\
\Leftrightarrow \cos 4x - \sin 4x = - \dfrac{{\sqrt 3 }}{2}\\
\Rightarrow \sin 4x - \cos 4x = \dfrac{{\sqrt 3 }}{2}\\
\Leftrightarrow \sqrt 2 \sin \left( {4x - \dfrac{\pi }{4}} \right) = \dfrac{{\sqrt 3 }}{2}\\
\Leftrightarrow \sin \left( {4x - \dfrac{\pi }{4}} \right) = \dfrac{{\sqrt 6 }}{4}\\
\Leftrightarrow \left[ \begin{array}{l}
4x - \dfrac{\pi }{4} = \arcsin \dfrac{{\sqrt 6 }}{4} + k2\pi \\
4x - \dfrac{\pi }{4} = \pi - \arcsin \dfrac{{\sqrt 6 }}{4} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{{16}} + \dfrac{{\arcsin \dfrac{{\sqrt 6 }}{4}}}{4} + \dfrac{{k\pi }}{2}\\
x = \dfrac{{5\pi }}{{16}} - \dfrac{{\arcsin \dfrac{{\sqrt 6 }}{4}}}{4} + \dfrac{{k\pi }}{2}
\end{array} \right.\left( {k \in \mathbb Z} \right)
\end{array}$
