NL-BXMO-TST 2022 Problem 3

Find all pairs $(p, q)$ of prime numbers such that $p(p^2 -p - 1) = q(2q + 3).$

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.