I've found the following statement at page 9 of Griffiths, Harris "Principles of Algebraic Geometry":
Proposition. If $R$ is a UFD and $u$, $v \in R[t]$ are relatively prime, then there exist relatively prime elements $\alpha$, $\beta \in R[t]$ and $\gamma \in R$, $\gamma \neq 0$ such that $\alpha u + \beta v = \gamma$.
How to prove this proposition? There are no hints in the book, so any help is very welcome.