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 $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such naturals is exactly $\dfrac{1}{\zeta{\left(2\right)}}$.