$f(x)=x^4+x+1$ is separable over $\Bbb F_p$ iff $p=229$

Question

I was asked to prove this in my homework, but when I tried looking at the roots of this polynomial over $\Bbb F_{229}$ I noticed $75$ is a double root of this polynomial and $f=(x-75)^2(x^2+150x+158)$. The factor $(x^2+150x+158)$ is ireducible over $\Bbb F_{229}$ (no root) so over a spliting field it decomposes as $f(x)=(x-75)^2(x-\alpha)(x+\alpha)$ where $\alpha^2+150\alpha +158=0$ is a root of $x^2+150x+158$ in the extension. How can this be separable over $\Bbb F_{229}$ if it has a double root in the base field?