We are given a polynomial $F(x)$. We are given enough coprime monic polynomials $p_{i}(x)$ that we are free to choose.
Let $F_{i}(x) \equiv F(x) \pmod {p_{i}(x)} \neq 0$.
Considering $F_i(x)$ as a polynomial in $\Bbb Z[x]$, say we are free to choose polynomials $g_{i}(x) \in \Bbb Z[x]$ of degree($g_{i}(x)$) < degree($p_{i}(x))$ such that:
$$F_{i}(x) + g_{i}(x) = s_{i}(x)t_{i}(x)$$
That is $F_{i}(x)$ when added with $g_{i}(x)$ factors into $s_{i}(x)t_{i}(x)$ with the added condition that all non-zero coefficients of $s_{i}(x)$,$t_{i}(x)$ are positive.
The question is if we use Chinese remainder theorem to find a polynomial $G(x)$ from $g_{i}(x)$ (given sufficiently many), then is it true that:
$$F(x) + G(x) = S(x)T(x)?$$
That is $F(x) + G(x)$ itself factors in $\Bbb Z[x]$ (Clearly the converse holds).
If this is always not possible under what extra conditions such a factorization may hold in $\Bbb Z[x]$?