CAPS 2007 Problem 1

Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$

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.