In $Z_5[x]$ , find all monic divisors of degree two for $2x^2+4x^3+3x^3+2x+4$

Question

In $\mathbb{Z}_5[x]$ , find all monic divisors of degree two for $2x^2+4x^3+3x^3+2x+4$

My attempt:

In $\mathbb{Z}_p[x]$ there are exactly $\frac{p^2-p}{2}$ monic irreducible polynomials of degree $2$

so in $\mathbb{Z}_5[x]$ there are 10 monic irreducible polynomials of degree $2$

But how to find all monic divisors of degree two for $2x^2+4x^3+3x^3+2x+4$ ?

Answer 1

Collect the terms in the polynomial to get $2x^3+2x^2+2x+4$ and rescale the polynomial by $3$ to get $x^3+x^2+x+2$ (call this $p$). Dividing $p$ will be equivalent to dividing the original.

One way to try and factor is to see what the roots are. By inspection, $1$ is the only root. So we factor the polynomial into $(x+4)$ times a cubic (which will also have no roots and therefore be irreducible). So any second degree factor that divides $p$ will either divide the cubic or will be a product $(x+4)$ and a factor that will divide the cubic contradicting that the cubic is irreducible. So there are no such second degree polynomials.