Find all pairs of integers $(a,b)$ such that $\frac{a^4-b+1}{ab}$ is an integer.
$b=1$ trivially gives infinitely many solutions as the expression becomes $a^3$. I am not able to find any more solutions. I tried Fermat's infinite descent to prove there are no solutions and got stuck... Also I have started reviewing Vieta's root jumping. Do I get some help on how to proceed... Thanks!