FKMO 2019 Problem 5

Find all pairs $(p,q)$ such that the equation $x^4+2px^2+qx+p^2-36=0$ has exactly $4$ integer roots(counting multiplicity).

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.