Using argument principle on $e^z + z$

Question

I want to use the argument principle to estimate the number of zeros of $e^z + z$ inside the rectangle with sides $y=2\pi n i, 2 \pi (n+1)i$ and $x = R, -R$ for $R$ large. But $\int \frac {e^z + 1}{e^z + z}dz$ doesn't seem so easy to integrate.

Answer 1

For $R$ large, the answer is $1$. See this by looking at the image of the rectangle under the mapping $z \rightarrow e^z + z$ which winds once around the origin.