VMO 2017 Problem 2

Is there an integer coefficients polynomial $P(x)$ satisfying $\begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2} P(1+\sqrt{5})=2+3\sqrt{5}\end{cases}$

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.