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*}