CGMO 2021 Problem 8

Let $m, n$ be positive integers, define: $f(x)=(x-1)(x^2-1)\cdots(x^m-1)$ , $g(x)=(x^{n+1}-1)(x^{n+2}-1)\cdots(x^{n+m}-1)$ .

Show that there exists a polynomial $h(x)$ of degree $mn$ such that $f(x)h(x)=g(x)$ , and its $mn+1$ coefficients are all positive integers.