Almost-prime values attained by a product of quadratic polynomials Let $F(x) = \prod_{i=1}^{k} (a_i x +b_i)$ be a product of $k$ linear polynomials, where $a_i,b_i$ are integers. Under very reasonable conditions, it is known that a constant $C_k$ exists with the following property: $F(n)$ is divisible by at most $C_k$ primes, for infinitely many $n$-s. (This is proven in Chapter 10.3 of `Sieve Methods' by Halberstam and Richert). In fact, $C_k$ is a very explicit constant, and one can require in addition that $F(n)$ will be squarefree and coprime to a given  $M$.

Question: Consider a polynomial $F(x) = \prod_{i=1}^{k} (a_i x^2 + b_i x +c_i)$, a product of $k$ quadratic polynomials. Suppose that the $k$ polynomials are pairwise coprime, and that $\gcd(F(1),F(2),F(3),\ldots)=1$. Is there a constant $D_k$ such that $F(n)$ is divisible by at most $D_k$ primes, for infinitely many $n$-s?

I am only aware about results on the case $k=1$, due to Iwaniec and Lemke-Oliver. I am interested in larger $k$, though. Even results for special $F$-s interest me, e.g. $F(x) = \prod_{i=1}^{k} ((a_ix)^2+1)$.
 A: A statement of this type follows from Selberg's sieve,
details are in Halberstam and Richert, Sieve methods, section 10.3 and 10.5.
(In the meantime there may be numerically somehwat stronger estimates, 
but the flavour might still be the same.)
Let me quote Theorem 10.11. (hence $r$ is your $C_k$ and $g$ is your $k$).
Let $F_1(n), \ldots, F_g(n)$ be distinct irreducible polynomials with integral coefficients, 
and let $F(n)$ denote their product. Also let $h_i$ denote the degree of $F_i$ and $G$ the degree of $F$. 
Let $\rho(p)$ denote the number of solutions of the congruence $F(n)\equiv 0 \bmod p$, 
and suppose that $\rho(p)<p$ for all $p$.
Then there exists a positive number $\delta=\delta(r,F)$ such that, as $x \rightarrow \infty$,
$$ | \{n:1\leq n \leq x, F(n)=P_r\}|\geq \delta \frac{x}{(\log x)^g} \left\{1+O_F\left(\frac{1}{\log \log x}\right)\right\} $$
for any natural number $r$, which satisfies either one of the two conditions
$r >G-1+ g \sum_{j=1}^g {\frac{1}{j}} + g \log \left(\frac{2G}{g}+ \frac{1}{g+1}\right)$, or ( a more complicated one).
Corollary 10.11.1 says that if the degree of all $F_i=h$ is constant
the condition on $r$ is (in a simplified form)
$r>g \log g +O_h(g)$.
