Prime polynomial values

Question

The number of prime values of the polynomial $n^3 − 10n^2 − 84n + 840$ where $n$ is an integer is..?

How do we do this? Is there some sort of specific method or formula that I can learn, if yes what is it called?

Answer 1

Taking Jyrki's hint:

$P(n) := n^3 − 10n^2 − 84n + 840=(n-10)(n^2-84)$

For $P(n)$ to be prime, therefore, we need the factors to be both positive or both negative, and one of them to be equal to $\pm1$. $|n^2-84|$ is never equal to one, so we only need to consider $n=9$ and $n=11$. $P(9)=3$ and $P(11)= 37$, so there are two values of $n$ with $P(n)$ prime.