A refined realization theorem in the context of the Schur-Szegő composition

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Every polynomial of the form P=(x+1)(xⁿ⁻¹+c₁xⁿ⁻²+ +cₙ₋₁) is representable as Schur-Szegő composition of n-1 polynomials of the form (x+1)ⁿ⁻¹(x+aᵢ), where the numbers aᵢ are unique up to permutation. We give necessary and sufficient conditions upon the possible values of the 8-vector whose components are the number of positive, zero, negative and complex roots of a real polynomial P and the number of positive, zero, negative and complex among the quantities aᵢ corresponding to P. A similar result is proved about entire functions of the form eˣR, where R is a polynomial.