Product of polynomials with negative coefficients Given $a \in \mathbb{Z}$ with $|a| &gt; 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)=\pm p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive coefficients? Is there a way to generate such polynomials?
$(x^2 - x + 1) (x+1) = x^3 + 1$

If you want strictly positive:

$(4x^2 - x + 1) (2x+1) = 8 x^3 + 2x^2 + x + 1$
Answered here:
by Gerry Myerson https://mathoverflow.net/questions/121568/on-reducible-polynomials