Exactly two of p,q,r have the same sign. Let p and q be those roots.

The quadriatic coefficient is 0, then r = -p-q.

The linear coefficient is -2017, which equals p*q+p*r+q*r. This simplifies to 2017 = p^2 + p*q + q^2. Since p and q are the same sign and integers, there are a finite set of candidates. Specifically both p and q are in the range [1,45], each solution in that range has a mirror counterpart in [-45,-1].

A quick computer search finds one distinct solution: p=41, q=7. Then |p|+|q|+|r| = 96.