I haven't really looked into the problem so I haven't been able to come up with a more elegant solution, but here's a go.
We have a 4th degree polynomial without a 3rd degree term so a possible factorization would be two second degree polynomials with opposite lineair terms.
Identifying coefficients gives us a 3x3 system
Solving yields
So we have the following factorization
Then it's just solving two quadratics, solutions are
Of course, these solutions for x depend on a, so I don't see why a negative solution would be impossible - for certain values of the parameter a.