Explicit upper bound for an average number of divisors of quadratic polynomials

Consider the divisor sum ${\sum }_{n\le N}\tau (n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve the maximal possible number of $D(-1)$-quadruples.