$F$ is a field, so by definition, $F$ is a commutative ring with unity in which every non-zero element is a unit. Then, $F[x]$ is a set of polynomials in which the coefficients come from $F$, so all of the non-zero coefficients have units in $F$.
This means we can have polynomials like $f=a_0+a_1x+a_2x^2+\cdots +a_nx^n$ where $a_k\in F,$ $0\le k\le n$.
If $F[x]$ were a field, every non-zero element would be a unit.
I'm not really sure where to go from here.