Polynomial Pell’s equation

Consider the polynomial Pell’s equation X 2 − D Y 2 = 1 X^2 -DY^2 = 1 , where D = A 2 + 2 C D = A^2 + 2C is a monic polynomial in Z [ x ] { Z}[x] and deg ⁡ C > deg ⁡ A {C} > {A} . Then for A , C ∈ Q [ x ] A, C ∈ { Q}[x] , deg ⁡ C > 2 {C} > 2 , and B = A / C ∈ Q [ x ] B = A/C ∈ { Q}[x] , a necessary and sufficient condition for the polynomial Pell’s equation to have a nontrivial solution in Z [ x ] { Z}[x] is obtained.