Solution:
Assume there exist polynomials $f$ and $g$ satisfying
$$f(x) g(x)=\left(x-a_{1}\right) \ldots\left(x-a_{n}\right)-1 .$$
Let $h(x)=f(x)+g(x)$. Now, for every $a_{i}$ we have $f\left(a_{i}\right) g\left(a_{i}\right)=-1$, so
$$h\left(a_{i}\right)=f\left(a_{i}\right)+g\left(a_{i}\right)=0 .$$
Thus $h(x)$ has at least $n$ distinct roots, so it has degree at least $n$. But since $\operatorname{deg} f+\operatorname{deg} g=n$, this can only occur if one of $f$ and $g$ is a constant polynomial, which is what we wanted to prove.