A sequence of d + 1 signs + and − beginning with a + is termed as a sign pattern (SP). We say that the real polynomial P := x d + d−1 j=0 a j x j , a j = 0, defines the SP σ := (+, sgn a d−1 ,. . ., sgn a 0). By Descartes' rule of signs, for the quantity of positive (resp. negative) roots of P , one has pos ≤ c (resp. neg ≤ p = d − c), where c and p are the numbers of sign changes and sign preservations in σ; the numbers c − pos and p − neg are even. We say that P realizes the SP σ with the pair (pos, neg). For SPs with c = 2, we give some sufficient conditions for the realizability of pairs (pos, neg) of the form (0, d − 2k), k = 1,. . ., [(d − 2)/2].