Suppose $f$ is not irreducible. Since $f(x)=x^4+x+4$ does not have any root in $\mathbb{F}\_5$, the only possible factorization can be $f(x)=(x^2+ax+b)(x^2+cx+d)$ for $a,b,c,d \in \mathbb{F}\_5$.

Now by comparing the coefficients one gets:

$$1. \,\,a+c \equiv 0 $$
$$2. \,\, b+d+ac \equiv0 $$
$$3. \,\,ad+bc \equiv 1 $$
$$4. \,\,bd \equiv 4 $$

From condition $1$ and $4$, one can find possible values of $a,b,c,d$ and show that none of them simultaneously satisfy $2$ and $3$. But this is a rather long process.

I came across this post [here](https://math.stackexchange.com/questions/1343450/how-can-i-prove-irreducibility-of-polynomial-over-a-finite-field) but Rabin's test is even more computationally difficult to do by hand. Is there any other way to tackle this without doing tediously long computations especially if one does not have access to computers?

I will appreciate any suggestions.