In $\mathbb{F}[x]$ the ideals are all principal . So the ideals are generated by monic polynomials. Now if $g(x)$ is a monic polynomial ,not contained in $(f(x))$ and which belongs to $\mathbb{F}[x]$ then $(g(x)/f(x))$ is an ideal in $\mathbb{F}[x]/(f(x))$ by the third isomorphism theorem.
This has been my attempt.Iam not sure whether the idea is clear and also I couldn't understand how to describe it in terms of factorization of $f(x)$.