Show that if $F$ is a field, $f(x) \in F[x]$, then $f(x) * F[x]$ is an ideal in $F[x]$, and every ideal of $F[x]$ is of this form.

Question

Show that if $F$ is a field, $f(x) \in F[x]$, then $f(x) \cdot F[x]$ is an ideal in $F[x]$, and every ideal of $F[x]$ is of this form.

I'm not sure how to start proving that. If anyone could give me a hint it would be highly appreciated.