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.

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