Let $a,b,a',b'$ be $r-\epsilon_1$ bit positive integers.
Let $c,d$ be $s+\epsilon_2$ bit positive integers.
Fix a pair $c,d$ and vary $a,b$ over all $r-\epsilon_1$ bit numbers. Do we have almost $2^{2r-2\epsilon_1}$ different $r+s+\delta$ bit integers each of form $ac+bd$ or $a'c+b'd$ for some fixed $\delta$?
Does this mean we have almost $2^{4r-4\epsilon_1}$ different $r+2s+\delta$ bit integers of form $c(ac+bd)+d(a'c+b'd)$ for some fixed $\delta$?
If we vary $c,d$ over $2^t$ different random pairs, do we have $2^{t+4r-4\epsilon_1}$ different integers of form $c(ac+bd)+d(a'c+b'd)$ for some fixed $\delta$?