Assume we know $f(x)\in\Bbb Z[x]$ of degree at least $4$ and we also know $r\in\Bbb Z$.
How do we find $\alpha(x),\beta(x)\in\Bbb Z[x]$ of any degree and $g(x),h(x)\in\Bbb Z[x]$ such that $$f(x)=(g(x)+\alpha(x) x(x-r))(h(x)+\beta(x) x(x-r))?$$
I need $deg(g(x)),deg(h(x))\geq2$.
Can we use LLL?