## Exhibit a pair of polynomials with a certain property

Exhibit two polynomials $p(x),q(x) \in \mathbb{Z}[x]$ of degree two such that every coefficient of $p(x)q(x)$ is even except for that of the linear term.

We desire $p(x)$ and $q(x)$ such that $\overline{p}\overline{q} \equiv x$ mod 2. Since $x$ is irreducible in $\mathbb{Z}/(2)[x]$, we must have (without loss of generality) $p(x) \equiv x$ and $q(x) \equiv 1$. Say $p(x) = 2x^2 + x + 2$ and $q(x) = 2x^2 + 2x + 1$; indeed $p(x)q(x) = 4x^4 + 6x^3 + 8x^2 + 5x + 2$ has the desired property.