I am good with all of this expect the last part about the quadratic factors. A pointer would be good.$\displaystyle \alpha$ is the complex root (with the smallest positive argument) of the equation $\displaystyle z^5 - 1 = 0$

show that $\displaystyle \alpha^4 = \alpha^*$. hence or otherwise obtain $\displaystyle z^5 - 1 = 0$ as a product of real linear and quadratic facto giving the coefficients in terms of integers and cosines.

Many Thanks, Bobak.