I have to find out how many roots the polynomial

has in the first quadrant.

Certainly, I need to use the argument principle, but I don't see how. Here's what I tried.

All zeroes of

lie in

which follows from applying the Cauchy bound. Suppose I can prove that

has no real and no imaginary roots. Then, since the function is holomorphic, I have to calculate

where

(integrals have to be calculated in the appropriate directions of course).

But to calculate those integrals, don't I need to know the zeroes of

Only then would I be able to integrate by residues, right? It's supposed to be very simple.