In this paper we study the robust D-stability of single-parameter polynomially-dependent matrices. D-stability of a matrix means that all the eigenvalues are in a prescribed open region, which is symmetric with respect to the real axis in the complex plane. We propose a method based on generalization of the stability feeler. By using this method, we can obtain complete D-stability intervals for a class of single-parameter polynomially-dependent matrices. This method does not require that a nominal matrix is stable. Moreover, we show an application to the analysis of robust D-stability in a physically motivated example.