BG-MO 1988 Problem 1

Find all real parameters $q$ for which there is a $p\in[0,1]$ such that the equation $x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0$ has four real roots.

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.