ZA-SAMO 2021 Problem 5

Determine all polynomials $a(x)$ , $b(x)$ , $c(x)$ , $d(x)$ with real coefficients satisfying the simultaneous equations \begin{align*} b(x) c(x) + a(x) d(x) & = 0 a(x) c(x) + (1 - x^2) b(x) d(x) & = x + 1. \end{align*}