Question 18
Find all polynomials $a(x), b(x), c(x), d(x)$ with real coefficients satisfying the simultaneous equations
$$
\begin{aligned}
b(x)c(x) + a(x)d(x) &= 0 \\
a(x)c(x) + (1 - x^2)b(x)d(x) &= x + 1
\end{aligned}
$$
$$
\begin{aligned}
b(x)c(x) + a(x)d(x) &= 0 \\
a(x)c(x) + (1 - x^2)b(x)d(x) &= x + 1
\end{aligned}
$$