Let $f(z) = 2z^{4}+5z^{2}$ and $g(z)=z^{4}+10z^{2}+1$. Prove that $f$ and $g$ have the same number of zeros inside the open unit disc as well as the same number of zeros outside the unit disc but inside the disc of radius $4$ centered at $0$.
Now, for the zeroes inside the unit disc we can apply Rouche's theorem we have $|g(z)-2f(z)|\le4<|2f(z)|\le14 $, when $|z|=1$ so $g$ and $f$ has the same number of zeroes inside the unit disc.
what about the other part of the question?
Thanks.