In the study of the Diophantine Equation $f(x)f(y) = f(z^2)$ where $f$ is quadratic, the computational proofs I have seen (for specific $f$) rely on Pell's Equation.
For example, if $f(t) = t^2+t+1$, then the equation becomes
$$(x^2+x+1)(y^2+y+1) = (z^4+z^2+1)$$
and via the substitution $y=x+2z$, and one more subsequent substitution, one eventually reaches the generalised Pell Equation
$$X^2-2Z^2=-7$$
which is known to have infinitely many positive integer solutions.
Has anyone come across other methods of showing infinitely many (nontrivial) integer or rational solutions for quadratic $f$ that do not rely on Pell Equations?
Thank you.