The result is true for any nonconstant polynomial $f(n) = a_m n^m + a_{m-1} n^{m-1} + \dots + a_0$ with integer coefficients. We may assume that $a_m > 0$. Thus there exists a positive integer $n_0$ such that $f(n)$ is positive and increasing on $(n_0, \infty)$.

It suffices to show that if for some $n_1 > n_0$, $f(n_1) = p_1^{r_1} \cdots p_k^{r_k}$ has exactly $k$ distinct prime factors, then for some $n_2 > n_1$, $f(n_2)$ has more than $k$ prime factors. Given such an $n_1$, let $n_2 = n_1 + p_1^{r_1+1} \cdots p_k^{r_k+1}$. Then
$$f(n_2) \equiv p_1^{r_1} \cdots p_k^{r_k} \pmod{p_1^{r_1+1} \cdots p_k^{r_k+1}}$$
Hence, for each $j$, $1 \le j \le k$, we have that $p_j^{r_j}$ divides $f(n_2)$ but $p_j^{r_j+1}$ does not divide $f(n_2)$. As $f(n_2) > f(n_1) = p_1^{r_1} \cdots p_k^{r_k}$, it follows that $f(n_2)$ must have at least $k+1$ prime factors.