Let $f \in \mathbb F_3[X]$ be reducible, degree 4 or 5 and no roots, then a monic irreducible polynomial exists of degree 2 dividing $f$.
I've proved that $X^2 + 1, X^2 + X + 2, X^2 + 2X + 2$ are the only monic irreducible polynomials of degree 2 in $\mathbb F_3[X]$.
In proving that one of these divide $f$, I know that $f = f_1 f_2$ for two polynomials of degree $\ge 1$. $\mathbb F_3[X]$ so every polynomial of degree $0$ is a unit. I also know that $\mathbb F_3[X]$ is a Euclidean domain, so there exists a unique factorization of $f = q_1 ... q_n$ of irreducible polynomials.
However how can I prove that either $f_1$ or $f_2$ is a monic irreducible polynomial of degree 2, or $q_j$ is such a polynomial in the unique factorization ?