I am trying to research ways to come up with better ways of factoring, which is why in any answer, I would not like factoring as that would defeat the whole point. As I was working, I came across this equation for one of my test values:$$y=x^2+12x+3$$To continue with my methods, I am asking what positive integer values of $k$ make this inequality true: $$N^2<k(x^2+12x+3)<(N+2)^2$$ where $N$ is a polynomial.
2026-03-28 02:22:13.1774664533
Diophantine Inequalities
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Having given a lot of attention to that quickly stated problem, I and perhaps others, wonder if the following is the intended question.
Given a polynomial $p(x)$ with positive integer coefficients,
for what positive integers $k$, is, for all real $x$,
$p(x)^2 < k(x^2 + 12x + 3) < (p(x) + 2)^2$?