SE-SKOLORNAS 1984 Problem 4

Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.

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.