Baltic Way 2022 · Problem 3

We call a two-variable polynomial $P(x,y)$ secretly one-variable if there exist polynomials $Q(x)$ and $R(x,y)$ such that $\deg(Q)\ge2$ and $P(x,y)=Q(R(x,y))$ (e.g. $x^2+1$ and $x^2y^2+1$ are secretly one-variable, but $xy+1$ is not).

Prove or disprove the following statement: If $P(x,y)$ is a polynomial such that both $P(x,y)$ and $P(x,y)+1$ can be written as the product of two non-constant polynomials, then $P$ is secretly one-variable.

Note: All polynomials are assumed to have real coefficients.