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 $k \in \mathbb{N}$.