Any 4th degree polynomial of the form:
can be factored into two quadratics of the form:
I'm not yet sure how useful this approach actually is, but I just plan to keep on working with it until I find out.
That comes straight out of the Fundamental Theorem of Algebra. Notice, however, that the factoring will usually not be unique.
Second of all, it is certainly true that $p + r = a,\ pr + q + s = b,\ ps + qr = c,\ and\ qs = d.$
This follows by equating coefficients after simplifying the product of the quadratics.
Now you have four equations in four unknowns, and you know there is least one valid solution. However, three equations are not linear, and I suspect you will find that, except in special cases, solving them involves solving the original quartic.