As I was browsing through the problems & solutions department of the American Mathematical Monthly the other day, I found out that, in his solution to problem E-2332 [1972, 87], which was published on page 77 of the first issue of vol. 80 of the AMM, Ernst Trost resorted to this trick and, what is more, added a reference to the aforementioned paper of his in Elemente der Mathematik for further applications of it.
The problem asked to find all solutions in positive integers of the equation $y^{3}+4y= z^{2}$. Trost's solution goes as follows:
We consider the more general eq. $ay^{3} +4a^{3}b^{4}y= z^{2}$, with $y , z>0$, where $a, b \in \mathbb{N}$. If $(y,z)$ is a solution, then the quadratic polynomial $P(t) =ay^{3}t^{2} -z^{2}t+4a^{3}b^{4}y$ has the rational zero $t=1$. Therefore, $z^{4}-(2aby)^{4}$ must be the square of an integer. Taking into account that $y>0$ and applying a result of Fermat (see, for instance: K. Conrad, "Proofs by descent", p. 7), we infer that $z=2aby$; it follows that the unique solution to the equation is $(y,z)= (2ab^{2},4a^{2}b^{3})$.