Let $F(x)$ be an irreducible polynomial with integer coefficients. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. For $F(t)=t$, we write $H_F(x, y,z)$ as $H(x, y,z)$ for simplicity. The estimate for $H(x, y,z)$ is classical and goes back to early work of Besicovitch and Erd\H os in the 1930s. In 2008, Kevin Ford determined the exact order of growth of $H(x,y,z)$ for all $x,y,z$. The corresponding estimate for a linear polynomial $F$ were obtained by Ford and his cooperators using Ford’s method. The study of $H_F(x,y,z)$ for a general polynomial of degree at least 2 began in connection with the problem of bounding from below the largest prime factor of $\prod_{n\le x} F(n)$. In this talk, we will first introduce the story on the investigation of $H(x, y,z)$. And then, we will show heuristic arguments for $H(x, y,z)$ and $H_F(x,y,z)$. Finally, we give some results on the estimate of $H_F(x, y, z)$ obtained by Ford and the speaker.