Summary: Let \(S \subset \mathbb R^{k + m}\) be a compact semi-algebraic set defined by \(P_{1}\geq 0,, P_{\ell}\geq 0\), where \(P_{i}\in \mathbb R[X_{1},\ldots, X_{k}, Y_{1},\ldots, Y_{m}]\), and deg\((P_{i})\leq 2, 1\leq i \leq \ell\). Let \(\pi\) denote the standard projection from \(\mathbb R^{k + m}\) onto \(\mathbb R^{m}\). We prove that for any \(q >0\), the sum of the first \(q\) Betti numbers of \(\pi (S)\) is bounded by \((k + m)^{O (q \ell)}\). We also present an algorithm for computing the first \(q\) Betti numbers of \(\pi (S)\), whose complexity is \((k+m)^{2^{O(q\ell)}}\). For fixed \(q\) and \(\ell\), both the bounds are polynomial in \(k + m\).