Rouche's Theorem and Zero Counting

Show that the function \(p(z)=z^4+2z^2-z+1\) has exactly one zero in each of the four quadrants.

Complete the following:

1. (10 pts) Prove Rouche’s theorem: Assume \(f\) and \(g\) are analytic functions inside and on a simple closed contour \(C\), and suppose \(|f(z)|>|g(z)|\) for all \(z\in C\). Then \(f\) and \(f+g\) have the same number of zeros inside \(C\).
2. (10 pts) Prove the following variant of the Argument Principle: Let \(C\) be a simple closed curve with counterclockwise orientation, and suppose \(f\) is analytic inside and on \(C\), except at \(M\) poles inside \(C\) (poles are counted with weight corresponding to their multiplicity). Let \(a\in \mathbb {C}\), and assume \(f(z)+a\) has no zeros on \(C\) and has \(N_a\) zeros inside \(C\) (again, counting according to multiplicity). Without citing the Argument Principle, prove that \[ \frac {1}{2\pi i}\oint _C \frac {f'(z)}{f(z)+a}\,dz=N_a-M. \]