I have a polynomial $f(z)=z^4+z^3-2z^2+2z+4$, and I want to find the number of roots in the first quadrant. I'm trying to use the argument principle (or Rouche), and I could try to make my contour the quarter circle, but I've having trouble because I can't justify that there are no roots on the real axis. Please give me some recommendations!
So now I understand why there are no roots on the contour; I have also justified that the integral on the arc goes to $2\pi i$ by normal limit considerations. However, I still am unsure how to figure out the arguments.