On the Density of naturals $n$ coprime to $\lfloor P(n)\rfloor$ for certain Classes of Polynomials

We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd (n,\lfloor P(n)\rfloor )=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such naturals is exactly ${\displaystyle \frac{1}{\zeta \left(2\right)}}$.