I am trying to factor the polynomial $2X^{10}+4X^5 +3$ into irreducible factors in $(\mathbb{Z} / 5\mathbb{Z})[X].$
I have already determined that the polynomial has no linear factor because $p(0) \neq 0,$ $p(1) \neq 0,$ $p(2) \neq 0,$ $p(3) \neq 0$ and $p(4) \neq 0.$
However, I am not sure how to continue.
Any help would be appreciated.
Use the fact that we have $(a+b)^5\equiv a^5+b^5\pmod 5$.
Specifically: $$2x^{10}+4x^5+3\equiv 2\,(x^{10}+2x^5+4)\equiv 2\,(x^2+2x+4)^5\pmod 5$$
It remains to show that $x^2+2x+4$ is irreducible $\pmod 5$ but that follows from what you have already done (or by direct computation).