This paper considers the problem of robust stability of a polynomial family whose coefficients are polynomial functions of the parameters of interests. The problem occurs in the design of a fixed order or fixed structure multivariable feedback controller, parametrized by a real design parameter vector, for a plant, containing a vector p of uncertain parameters. The characteristic polynomials of such systems often contain coefficients which depend polynomially on and p. Using results on sign-definite decomposition, a new stability test is developed that gives a sufficient condition for Hurwitz stability of the family of closed loop systems that result when and p vary over prescribed boxes. This test is reminiscent of Kharitonov's Theorem, even though the family of polynomials considered here is certainly not restricted to be interval or even convex. Moreover the test does reduce to Kharitonov's Theorem for the special case of interval polynomials. Using this criterion recursively and modularly, sets of controllers that stabilize the family of uncertain plants are determined.