The answer of VA is quite simple to understand. In fact the result is of local nature.
Proposition: Let $R$ be a noetherian (commutative unitary) ring, let $B$ be a flat noetherian $R$-algebra, and let $f\in B$ be an element such that for any maximal ideal $m$ of $R$, the image of $f$ in $B/mB=B\otimes_R R/m$ is a regular element. Then $B/fB$ is flat over $R$.
This can be found for example in Matsumura, page 151, (20.F) (taking $M=B$). It is also in Milne's "Etale cohomology", first chapter.
To apply to your concrete situation, $B$ is a polynomial ring over $R$, and $f$ is a polynomial which is non-zero modulo $m$ (cf. explanation of VA), so it is regular modulo $m$ because $B/mB$ is an integral domain. Therefore $B/fB$ is flat over $R$.