In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone.

Assume that $\mathbb F_p$ is the field with $p$ elements, and that we choose an integer $a$ with $1< a < 4p$, such that $a$ be coprime to $p(p-1)$ (for example, $a = 2p - 1$ is suitable). Let $t$ be transcendental over $\mathbb F_p$, and set $K = \mathbb F_p(t)$. Consider the polynomial $$P(X) = X^{4p} +t^a$$ in $K[X]$.

Then for every $f\in \mathbb F_p[t]$, $P(f)$ is reducible in $\mathbb F_p[t]$.

I would like a direct proof of this result. Any idea?