I know we need to use the argument principle to solve this, but I don't know how to use this.
Argument Principle states: $$ \text{number of zeros}=\frac{1}{2\pi i}\int_{\partial\Omega}\frac{f'(z)}{f(z)}dz. $$ Here $$ \partial\Omega=\{z=iy, -R\leq y\leq R\}\cup\{ z=Re^{i\theta},-\frac{\pi}{2}\leq \theta\leq\frac{\pi}{2}\}. $$ It claims that $$ \int_{-iR}^{iR}\frac{f'(z)}{f(z)}dz=0. $$
I need to find the number of zeros in the first quadrant. This is a problem for my qualifying exam practice. I found it in Nakhle's complex analysis for applications, and I am not understanding how to apply this theorem. Any ideas on how to proceed or solve this would be greatly appreciated.
The function is $$ f(z) = z^5 + z^4 + 4z^3 + 10z^2 + 9$$