Almost primes in various settings

Let $k\ge 3$ and let $L_i(n)=A_{i}n+B_i$ be some linear forms such that $A_i$ and $B_i$ are integers. Define $\mathcal{P}(n)={\prod }_{i=1}^k{L}_i(n)$. For each $k$ it is known that $\mathrm{\Omega }(\mathcal{P}(n))\le {\rho }_k$ infinitely often for some integer ${\rho }_k$. We improve the possible values of ${\rho }_k$ for $4\le k\le 10$ assuming $GEH$. We also show that we can take ${\rho }_5=14$ unconditionally. As a by-product of our approach we reprove the ${\rho }_3=7$ result which was previously obtained by Maynard who used techniques specifically designed for this case.