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Secret one variable polynomials

Source: Baltic Way 2022, Problem 3

November 12, 2022
algebrapolynomial

Problem Statement

We call a two-variable polynomial P(x,y)P(x, y) secretly one-variable, if there exist polynomials Q(x)Q(x) and R(x,y)R(x, y) such that deg⁡(Q)≥2\deg(Q) \ge 2 and P(x,y)=Q(R(x,y))P(x, y) = Q(R(x, y)) (e.g. x2+1x^2 + 1 and x2y2+1x^2y^2 +1 are secretly one-variable, but xy+1xy + 1 is not).
Prove or disprove the following statement: If P(x,y)P(x, y) is a polynomial such that both P(x,y)P(x, y) and P(x,y)+1P(x, y) + 1 can be written as the product of two non-constant polynomials, then PP is secretly one-variable.
Note: All polynomials are assumed to have real coefficients.
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