I am struggling to understand the solution below. I understand how to apply Rouches theorem when showing that there are a certain number of zeroes in a circle / annulus.
In this example (where they ask to show that there is a zero in each quadrant, I am not so sure. I dont really understand why they use inequalities with the real and imaginary parts.
I also dont understand why you can say that since since each of the roots of $z^4+1$ lie in a separate quadrant, then they must lie in separate quadrants of $z^4+z+1$ also.