A Q-derived polynomial is a univariate polynomial, defined over
the rationals, with the property that its zeros, and those of all
its derivatives are rational numbers. There is a conjecture that
says that Q-derived polynomials of degree 4 with distinct
roots for themselves and all their derivatives do not exist. We
are not aware of a deeper reason for their non-existence than the
fact that so far no such polynomials have been found. In this
paper an outline is given of a direct approach to the problem of
constructing polynomials with such properties. Although no
Q-derived polynomial of degree 4 with distinct zeros for
itself and all its derivatives was discovered, in the process we
came across two infinite families of elliptic curves with
interesting properties. Moreover, we construct some K-derived
polynomials of degree 4 with distinct zeros for itself and all
its derivatives for a few real quadratic number fields K of
small discriminant