ANSWER:
C.
[tex]x=\pi,x=\frac{2\pi}{3},x=\frac{4\pi}{3}[/tex]STEP-BY-STEP EXPLANATION:
We have the following function:
[tex]2cos^2x+3cosx\: +1\: =\: 0[/tex]Using the substitution method, we can calculate the value of x, like this:
[tex]\begin{gathered} u=\cos x \\ \text{ therefore:} \\ 2u^2+3u+1=0 \\ 3u=2u+u \\ 2u^2+2u+u+1=0 \\ 2u(u+1)+u+1=0 \\ (u+1)(2u+1)=0 \\ u+1=0\rightarrow u=-1 \\ 2u+1=0\rightarrow2u=-1\rightarrow u=-\frac{1}{2} \\ \text{ replacing:} \\ \cos x=-1\rightarrow x=\cos ^{-1}(-1)\rightarrow x=\pi \\ \cos x=-\frac{1}{2}\rightarrow x=\cos ^{-1}(-\frac{1}{2})\rightarrow x=\frac{2\pi}{3},\frac{4\pi}{3} \end{gathered}[/tex]