Showing a degree 6 polynomial is irreducible over $\mathbb{Z}_5$

Question

Consider the polynomial $g(x) = x^6 + 2x^5 + 4x^4 + 3x^3 + x^2 + 2x + 3$. I am trying to show that this polynomial is irreducible in $\mathbb{Z}_5[x]$. So far I have shown that $g$ has no roots in $\mathbb{Z}_5$ and hence has no linear factors over $\mathbb{Z}_5$. My next step is to show that $g$ has no irreducible factors in $\mathbb{Z}_5[x]$ of degree $2$ or $3$. Is there a more efficient way of proving this, besides enumerating all of the irreducible polynomials of degree 2 or 3 in $\mathbb{Z}_5[x]$ and then checking that none of them divides $g$ in $\mathbb{Z}_5[x]$?