Counting the zeros when a polynomial is perturbed by an exponential
Take $U$ the domain bounded by $|z|=1, |z|=2, \Re z=0$ and note that for $|z| \le 1, \Re z \le 0, |f(z)| \ge 8-2>0$ while for $|z| \ge 2, \Re z \le 0, |f(z)| \ge 32-8-1>0$ so all its left plane zeroes are inside $U$
But now on the boundary of $U$ we have $|e^{\gamma z}| \le 1$ while on the interior circle part $|z^5-8| \ge 7$, on the exterior circle part $|z^5-8| \ge 24$ and on the imaginary axis $z^5$ is imaginary so $|z^5-8| \ge 8=|\Re (z^5-8)|$ so by Rouche $f$ and $z^5-8$ have the same number of zeroes in $U$, namely $2$