44. (ROM 3) Suppose $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q)$.
(a) Find the sum $S(x, n)=\sum(x+p)(x+q)$, where $p$ and $q$ take values from 1 to $n$.
(b) Do there exist integer values of $x$ for which $S(x, n)=0$ ?
44. (a) $S(x, n)=n(n-1)\left[x^{2}+(n+1) x+(n+1)(3 n+2) / 12\right]$.
(b) It is easy to see that the equation $S(x, n)=0$ has two roots $x_{1,2}=$ $(-(n+1) \pm \sqrt{(n+1) / 3}) / 2$. They are integers if and only if $n=$ $3 k^{2}-1$ for some $k \in \mathbb{N}$.