Question 83
Find all pairs $(P, Q)$ of polynomials with real coefficients such that
$$
\frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}
$$
for infinitely many $x \in \mathbb{R}$.
$$
\frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}
$$
for infinitely many $x \in \mathbb{R}$.