  Given a natural number $n \geq 4$ we show that there exists infinitely many
polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i)
$f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no
rational roots, and (iii) every proper divisor of $f_{n}(x)$ fails to have root
modulo some positive integer. We exhibit a process to explicitly construct such
$f_{n}$ and this process demonstrates that the set of natural numbers $a_{n}$,
such that the polynomial $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ satisfies
the properties (i), (ii) and (iii), is of positive asymptotic density in
$\mathbb{N}$.
