DE-BUNDESWETTBEWERB 2001 Problem 6

For each $n \in \mathbb{N}$ we have two numbers $p_n, q_n$ with the following property: For exactly $n$ distinct integer numbers $x$ the number $$x^2 + p_n \cdot x + q_n$$ is the square of a natural number. (Note the definition of natural numbers includes the zero here.)