The disconnectedness of certain sets defined after uni-variate polynomials

We consider the set of monic real uni-variate polynomials of a given degree d with non-vanishing coefficients, with given signs of the coefficients and with given quantities pos of their positive and neg of their negative roots (all roots are distinct). For d >= 6 and for signs of the coefficients (+, -, +, +, ... , +, +, -, +), we prove that the set of such polynomials having two positive, d - 4 negative and two complex conjugate roots, is not connected. For pos + neg <= 3 and for any d, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of pos and neg.