Why is the argument principle not working?

$\oint \frac{2z+1}{z^2+z} dz$, where $|z|=2$.

It seems a pretty simple application of the argument principle

THe zeros: $$z = -1/2$$

The poles: $$z = 0,-1$$

Assim, temos que $$N0 - NP = -1$$

Logo, $\oint \frac{2z+1}{z^2+z} dz = -2 \pi i $

Which is obviously wrong: A fast application of cauchy theorem + residues theorem show that the integral if $4 \pi i$. But if so, where is my error? Since is the first time i am seeing the argument principle, i think i am missing something.

Solution 1:

Here, you are applying the argument principle to $f(z)=z^2+z$, which has two zeros and zero poles in that region bounded by the loop. So, it is natural that the answer is $4\pi i$.