How to solve $2 \sin^2(x)+\cos^2(x)+\cos(x)=0$ I need to solve the following equation:

$$2 \sin^2(x)+\cos^2(x)+\cos(x)=0$$

I think that $\sin^2(x)+\cos^2(x) = 1$ is helpful, but I don't know how to apply it here.
You can rewrite the equation to

$$2(1-\cos^2x) + \cos^2 x + \cos x = 0$$

Now, introduce a new variable: $y=\cos x$ and first solve for $y$.

Then, for each solution $y$, every $x$ that solves the equation $\cos x = y$ solves the original equation.
$2sen^2(x)+cos^2(x)+cos(x)=sen^2(x)+sen^2(x)+cos^2(x)+cos(x)=sen^2(x)+cos(x)+1=(1-cos^2(x))+cos(x)+1=2+cos(x)-cos^2(x)=0$. 

This last equation is simpler.

Take $z:=cos(x)$ then $z^2-z-2=0$