Let $k$ be a even positive integer. Now, consider the polynomial $$ p(x)=x^k-px^{k-1}-qx^{k-2}-x^{k-3}-\cdots -x-1, $$ with $p$ and $q$ integers satisfying $q-1>p\geq 1$.
How to prove that this polynomial has k-2 roots inside the unit circle (there are two roots outside this circle by using the Descarte sign rule and the intermediate value theorem).
Any suggestion? I tried to use Rouché theorem, but without success.