Consider the behavior of non-constant integer polynomials as \( x \) grows large. How does \( |f(x)| \) behave asymptotically, and what does this imply about its divisibility properties?

Recall that for any integer polynomial \( f(x) \), there exists an integer \( c \) such that \( f(x) \equiv c \pmod{m} \) for some modulus \( m \). How can this be used to construct infinitely many \( x \) where \( |f(x)| \) is divisible by a fixed integer?

Use the Chinese Remainder Theorem to construct an infinite sequence of integers \( x \) such that \( f(x) \) is divisible by a product of distinct primes, ensuring \( |f(x)| \) is composite for each \( x \) in the sequence.