Determine all real polynomials $P$ of degree at most $22$ for which $$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$for all $k \in \{1, 2, 3, . . . , 21, 22\}$.
Source: Flanders Math Olympiad 2022 p4
Tags: algebra, polynomial
Determine all real polynomials $P$ of degree at most $22$ for which $$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$for all $k \in \{1, 2, 3, . . . , 21, 22\}$.
