Guidance on a Complex Analysis question My homework question:  Show that all zeros of $$p(z)=z^4 + 6z + 3$$ lie in the circle of radius $2$ centered at the origin.
I know $p(z)$ has a zero-count of $4$ by using the  Fundamental Theorem of Algebra. Then using the Local Representation Theorem the $$\int \frac{n}{z+a} = 4(2 \pi i).$$ I am assuming $a=0$ since we are centered at the origin. I apologize for my lack of math-type.  What does $$= 8 \pi i$$ mean?  Am I going around the unit circle $4$ times?  Or is it even relevant to my final answer.  Which I am assuming is finding the coordinates to the $4$ singularities. I have always looked for my singularities in the values that make the denominator zero, but in this question my denominator is $z$.  $z=0$ doesn't seem right.  So the question is, am I suppose to factor the polynomial $z^4 + 6z + 3$ to find the zeros?
Thanks

Hint: This kind of questions are usually handled using Rouche's Theorem. I suggest you look it up in the wikipedia article, where you can see an example of its usage. Also here's an example.
The key is choosing wisely another function $f(z)$ with which to compare in the inequality in Rouche's theorem and such that you can easily decide how many zeroes does $f(z)$ have inside the region you are considering, which in your case is the circle $|z| < 2$.
About your other question, you don't need to factor the polynomial in order to answer this.
