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We prove that every Diophantine quadruple in $\mathbb{R}[X]$ is regular. Moreprecisely, we prove that if $\{a, b, c, d\}$ is a set of four non-zeropolynomials from $\mathbb{R}[X]$, not all constant, such that the product ofany two of its distinct elements increased by $1$ is a square of a polynomialfrom $\mathbb{R}[X]$, then $$(a+b-c-d)^2=4(ab+1)(cd+1).$$  One consequence of this result is that there does not exist a set of fournon-zero polynomials from $\mathbb{Z}[X]$, not all constant, such that aproduct of any two of them increased by a positive integer $n$, which is not aperfect square, is a square of a polynomial from $\mathbb{Z}[X]$. Our resultalso implies that there does not exist a set of five non-zero polynomials from$\mathbb{Z}[X]$, not all constant, such that a product of any two of themincreased by a positive integer $n$, which is a perfect square, is a square ofa polynomial from $\mathbb{Z}[X]$.

A polynomial variant of a problem of Diophantus and its consequences