As relayed in this question of mine (which is more general in scope), I believe I have found a relatively easy, and completely elementary, way to show that the equation $$48X^4 + 12X^2+1 = Y^2$$ has only the trivial integer solution $(x,y) = (0, \pm 1)$. I did it by reducing it, via fairly trivial means, to the equation $$w(w+1)=12z^2(16z^2+1),$$ setting \begin{align} w &= ab, & 12z^2 &= ac, \\ w+1 &= cd, & 16z^2+1 &= bd, \end{align} and then manipulating those until I could show that $a=0$ (implying $w=z=0$, leading back to $x=0$, as claimed).
Now I'm wondering if anyone else has a better elementary solution.