Quartic Integral Polynomial Pell Equations

In this paper we classify all monic, quartic, polynomials $d(x)\in \mathbb{Z}[x]$ for which the Pell equation $$f(x{)}^2-d(x)g(x{)}^2=1$$ has a non-trivial solution with $f(x),g(x)\in \mathbb{Z}[x]$.