If \(R\) is a commutative ring with unity, and \(I\neq 0\) its proper ideal, then denote by \(R[It]=\bigoplus_{n\geq 0}(I^nt^n)\) the Rees algebra associated with \(R\), relative \(I\); for \(a,b\in R\) denote by \((I^2(t^2+at+b))\) the contraction to \(R[It]\) of the ideal generated by \(t^2+at+b\) in \(R[t]\). V. Barucci and present authors introduced [Commun. Algebra 43, No. 1, 130–142 (2015; Zbl 1327.13087)] \(R[It]/(I^2(t^2+at+b))=R(I)_{a,b}\). This paper is a continuation of the authors’ work in this area. They look into relationships between the prime spectrum of \(R\) and prime spectrum of \(R(I)_{a,b}\), giving its complete description. Localizations at a prime ideal are also described. The results are in terms of conditions on \(t^2+at+b\), such as whether it is irreducible or not. This enables them to characterize Cohen-Macaulay and Gorenstein properties. The authors also look into conditions that ensure that \(R(I)_{a,b}\) is a domain, that it is reduced, quasi-Gorenstein, as well as for it to satisfy Serre’s conditions.