While reading an article by Dan Carmon on Square-free values of large polynomials over the rational function field, I tried to prove that square-free polynomials of degree $1$ and $2$ have a positive density. This I was able to do with some help of the article Squarefree values of quadratic polynomials by Zeev Rudnick
My quesion Why doesn't this strategy work for polynomials of degree $3$ and why do we need the ABC-conjecture here, i.e., why can't we use the Sieve of Eratosthenes for polynomials of degree 3? I suppose it has something to do with the fact that when looking at polynomials of degree $1$, multiples of primes are evenly spaced. Any thoughts?