Note on the number of divisors of reducible quadratic polynomials

In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum \(\sum_{n \leq x} d( n (n+v))\) where \(v\) is a fixed integer and \(d(n)\) denotes the number of divisors of \(n\). We reprove her result by following a suggestion of Hooley, namely investigating the relationship between this sum and the well-known sum \(\sum_{n \leq x} d( n ) d (n+v)\). As such, we are able to furnish additional terms in the asymptotic formula.