abstract algebra - Show that $H = mathbb{Z}_5[x]/langle x^4+3x^3+x+4rangle$ is not a field.

So I am looking over old exams in abstract algebra and I came across this question which seems to be a mistake. (Neither the original teacher who wrote it, nor my own teacher are available to answer)

Let $H = \mathbb{Z}_5[x]/\langle x^4+3x^3+x+4\rangle$. Show that $H$ is not a field.

Letting $p(x) = x^4+3x^3+x+4 $, we can see that $p(x)$ has no solutions in $\mathbb{Z}_5$. Therefore $p(x)$ is irreducible over $\mathbb{Z}_5$, and thus $\langle p(x)\rangle$ is a maximal ideal. Now the factor group of a ring and a maximal ideal is a field. Thus H must be a field.

Am I wrong here, and if so how?

Answer

By the Berlekamp algorithm we obtain
$$
x^4+3x^3+x+4=(x^2 + 4x + 2)^2
$$
over $\mathbb{F}_5$. Hence the quotient is not a field, because it has zero divisors.