$4(sin^4x + cos^4x) + \sqrt{3}sin4x = 2 \Leftrightarrow 4(1-2sin^2xcos^2x)+\sqrt{3}sin4x=2$
$ \Leftrightarrow -8sin^2xcos^2x+\sqrt{3}sin4x+2=0 \Leftrightarrow-8sin^2xcos^2x+2\sqrt{3}sin2xcos2x+2=0$
$ \Leftrightarrow -4sin^2xcos^2x+\sqrt{3}sin2xcos2x+1=0 \Leftrightarrow 1-sin^22x+\sqrt{3}sin2xcos2x=0 \Leftrightarrow cos2x(cos2x+\sqrt{3}sin2x)=0 $