Solutions of polynomial Pell's equation

Let D=F2+2G be a monic quartic polynomial in Z[x], where . Then for F/G∈Q[x], a necessary and sufficient condition for the solution of the polynomial Pell's equation X2−DY2=1 in Z[x] has been shown. Also, the polynomial Pell's equation X2−DY2=1 has nontrivial solutions X,Y∈Q[x] if and only if the values of period of the continued fraction of are 2, 4, 6, 8, 10, 14, 18, and 22 has been shown. In this paper, for the period of the continued fraction of is 4, we show that the polynomial Pell's equation has no nontrivial solutions X,Y∈Z[x].