So let $R$ be an integral domain and $K$ its fraction field. Let $f,g,h\in K[x]$
be 3 monic polynomials such that $f=gh$. So I would like to have a simple example
of a ring $R$ for which one has that $f,g\in R[x]$ but $h\notin R[x]$. 


P.S. May be working with an non-maximal order of a Dedekind ring is good enough. Nevertheless
I could not come up with such an example 



quotient of integral polynomials not being integral
We show that $h$ must be in $R[x]$. Suppose that we have polynomials


$$g = x^n + a\_1x^{n-1} + ... + a\_n \in R[x]$$
$$h = x^m +b\_1x^{m-1} + ... + b\_m \in K[x]$$
 such that $f = gh \in R[x]$ but $h \notin R[x]$. Let $r:= \min \{i \mid b\_i \notin R\}$. Since $f \in R[x]$ we have $b\_r + a\_1b\_{r-1} + ... + a\_{r-1}b\_1 + a\_r \in R$, so $b\_r \in R$. It is a contradiction. 


