Working with the famous Baker-Davenport system of simultaneous Pell equations \begin{align} 3x^2-2 &= y^2, & 8x^2-7 &= z^2, \qquad(\star) \end{align} I am left, after a series of substitutions and reductions, with the equation $$ u(u+1)\left(96u(u+1)+11\right) = v(v+1). \qquad(\dagger) $$ The point is now to prove that the only solutions with $u \ge 0$ are $u=0$ and $u=5$, yielding, respectively, the two known (and only) solutions $x=1$ and $x=11$ in ($\star$).

Are there any ways of attacking the formulation ($\dagger$), different from those which would naturally be applied to ($\star$), which might lead to a relatively straight-forward proof?

Thanks,
Kieren.