8.357. $4 \sin ^{4} x+\cos 4 x=1+12 \cos ^{4} x$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

8.357. $4 \sin ^{4} x+\cos 4 x=1+12 \cos ^{4} x$.
## Solution.

Rewrite the equation as

$4\left(\sin ^{2} x\right)^{2}+\cos 2(2 x)-1-12\left(\cos ^{2} x\right)^{2}=0 \Leftrightarrow$

$\Leftrightarrow(1-\cos 2 x)^{2}+2 \cos ^{2} 2 x-1-1-3(1+\cos 2 x)^{2}=0 \Leftrightarrow \cos 2 x=-\frac{1}{2}$,

$2 x= \pm \frac{2}{3} \pi+2 \pi k, x= \pm \frac{\pi}{3}+\pi k=\frac{\pi}{3}(3 k \pm 1), k \in Z$.

Answer: $x=\frac{\pi}{3}(3 k \pm 1), k \in Z$.