Answer
$0,\frac{2\pi}{3},\pi,\frac{4\pi}{3}$
Work Step by Step
Step 1. Factor the equation as $cos^2(x)(2cos(x)+1)-(2cos(x)+1)=(2cos(x)+1)(cos^2(x)-1)=0$ and use $cos^2(x)-1=-sin^2x$; we have $(2cos(x)+1)(sin^2(x))=0$ which gives $sin(x)=0$ and $cos(x)=-\frac{1}{2}$
Step 2. For $sin(x)=0$, the solutions in $[0,2\pi)$ are $x=0,\pi$
Step 3. For $cos(x)=-\frac{1}{2}$, we can find the solutions in $[0,2\pi)$ as $x=\frac{2\pi}{3},\frac{4\pi}{3}$
Step 4. The solutions to the original equation are $0,\frac{2\pi}{3},\pi,\frac{4\pi}{3}$
