If $a=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7},\alpha =a+a^2+a^4$  and  $\beta =a^3+a^5+a^6$  then $\alpha ,\beta$  are the roots of the equation

$a=Cos\frac{2\pi }{7}+iSin\frac{2\pi }{7}=e^{i\frac{2\pi }{7}}$
$\Rightarrow a^7=e^{i2\pi }=Cos2\pi +i\sin 2\pi =1$
 $$\begin{gathered} \Rightarrow a^7=1 \\ \alpha +\beta =a+a^2+a^3+a^4+a^5+a^6 \\ =\frac{a\left(a^6-1\right)}{a-1}=\frac{a^7-a}{a-1}=\frac{1-a}{a-1}=-1 \end{gathered}$$

$\alpha \beta =\left(a+a^2+a^4\right)\left(a^3+a^5+a^6\right)$
$=a+a^2+a^3+a^4+a^5+a^6+3a^7$

$=-1+3=2$

$\alpha +\beta =-1, \alpha \beta =2$

$QE\Rightarrow x^2-x\left(-1\right)+2=0$

$\Rightarrow x^2+x+2=0$