$2\sin^2 x + 4\sin^2 x\cos^2 x = 2$
$\sin^2 x(1 + 2\cos^2 x) = 1$
Let $t = \sin^2 x$: $t(3-2t) = 1 \Rightarrow 2t^2 - 3t + 1 = 0$
$(2t-1)(t-1) = 0 \Rightarrow t = \frac{1}{2}$ or $t = 1$
$\sin^2 x = \frac{1}{2}$: 4 solutions; $\sin^2 x = 1$: 2 solutions. Total $= 6$