jmc

Because the quadratic has two distinct integer roots, we know that it can be factored as $$(x+r)(x+s),$$ where $r$ and $s$ are positive integers. Expanding this product gives $x^2+(r+s)x+rs$, and comparing this to the given quadratic tells us that $rs=36$. So, we consider all the pairs of distinct integers that multiply to 36, and we compute their sum in each case: $$\begin{array}{ccc}r& s& r+s\\ 1& 36& 37\\ 2& 18& 20\\ 3& 12& 15\\ 4& 9& 13\end{array}$$ Summing the entries in the final column gives us a total of $\boxed{85}$.