Answer
$x=\{2.4981,4.7124\}$
Work Step by Step
$2cos(x)+sin(x)+1=0$
$[sin(x)+1]^2=[-2cos(x)]^2$
$sin^2(x)+2sin(x)+1=4cos^2(x)$
$sin^2(x)+2sin(x)+1-4cos^2(x)=0$
$sin^2(x)+2sin(x)+1-4[1-sin^2(x)]=0$
$sin^2(x)+2sin(x)+1-4+4sin^2(x)=0$
$5sin^2(x)+2sin(x)-3=0$
$sin(x)=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{-(2) \pm \sqrt{(2)^2 - 4 (5)(-3)}}{2(5)}=\frac{3}{5},-1$
$sin(x)=\frac{3}{5}=0.6$
$x=sin^{-1}(\frac{3}{5})$
$x=2.4981$
$sin(x)=-1=\;\;\;\;$
$x=sin^{-1}(-1)$
$x=4.7124$
$x=\{2.4981,4.7124\}$