How do I solve the equation given below for values for $\epsilon$?
$ \sum_{j=1}^{n-1} (2{\alpha_{j}}+1) ~~ \prod_{k=1,~k\neq j}^{n-1} \bigg(({\alpha_{k}}-1){\Big((2{\alpha_{k}}+3) + {\epsilon}(2{\alpha_{k}}+1)\Big)}^{2}\bigg) = 0 $
where $\alpha_i = \cos{\dfrac{2\pi i}{n}}$, where $n$ is an even integer and $n \geqslant 8$. ($j$ is NOT the imaginary unit).