Check to see if $f$ has “roots” or find an $\alpha$ where $f(\alpha)=0$.
If you find a root then it means that there is a factor $x-\alpha$!
If there are no roots, then it means that it is irreducible and you are done!
Let $f(x)=ax^3+bx^2+c$
We know that if there is a rational root it must be of the form
$$ \frac{r} s\ st.\ s\mid a\text{ and } r\mid c $$
If a root $\alpha$ is found, then we either have a factor $x-\alpha$ or it is irreducible