Irreducible Polynomials in UFD and corresponding Quotient Field

Hello,

"Let $D$ be a UFD and let $F$ be its quotient field. Further let
$f$ be a primitive polynomial of positive degree in $Dleft[xright]$.
From this it follows that that $f$ is irreducible in $Dleft[xright]$ if and only
if $f$ is irreducible in $Fleft[xright]$."

I've shown the forward direction, i.e. irreducible in UFD $Rightarrow$ irreducible in quotient field, but am struggling to understand how the converse direction would go.

In particular, it strikes me that it might be productive to try and show that $f$ is prime in the UFD since this is equivalent to irreducibility, but I don't know how to show this. Also, $f$ should have no "denominators" in $Fleft[xright]$ since it's also in $Dleft[xright]$.

Any advice on how to proceed?