Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with factors not congruent to $1 \mod g(x)$?

If so, what is the best way to find one such $h(x) \in \mathbb{Z}[x]$?

Let degree of $f(x),g(x)$ be bounded by $n$ (with degree of $g(x)$ strictly less than degree of $f(x)$) and maximum size of coefficients of $f(x),g(x)$ be $2^{s}$. Is there a $(ns)^{c}$ or $(n2^{s})^{c}$ algorithm to find such an $h(x)$ (if such a polynomial exists) where $c \in \mathbb{N}$ is fixed?

In general, is there a way to define the probability that $f(x) + g(x)h(x)$ is reducible into factors not congruent to $1 \mod g(x)$? It is likely this probability asymptotically goes to zero in terms of $n$ and $s$. However how fast does it go to zero? Does it go to zero asymptotically as $\frac{1}{(ns)^{c_{1}}}$ or $\frac{1}{(n2^{s})^{c_{1}}}$ for some positive constant $c_{1}$?