see this link Rouché's theorem - Wikipedia, the free encyclopedia

First by the fundemental theorem of Algebra f has 5 zero's including multiplicity and Consider the Unit Disk.

Define and note that g(z) has 5 zero's in the unit disk (i.e 0 is a root of multiplicity 5)

Now lets check Rouche's theorem

on the boundry of the unit disk

and

so then

on the boundry so they gave the same number of zero's on the unit disk.

So f has 5 zero's inside the unit disk. Yay