the first part is straightforward. for the roots i get $\displaystyle \alpha^0 , \alpha^1 , \alpha^2 , \alpha^3 , \alpha^4 , \alpha^5 , \alpha^6$.

For the quadratic I guessed got the root as the complex conjugate which is $\displaystyle \alpha^{-1} + \alpha^{-2} \alpha^{-4}$

i get that $\displaystyle A= - 2 \left( \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{8 \pi}{7} \right)$

and B is the product of the roots, but I am having trouble showing the required results.