Show that $\frac{(x+2)^p-2^p}{x}$ is irreductible in $\mathbb{Z}[x]$ for $p$ an odd prime number

Question

Show that $\frac{(x+2)^p-2^p}{x}$ is irreductible in $\mathbb{Z}[x]$ for $p$ an odd prime number.

I think I have to use the Eisenstein's criterion, but I don't know how to use it. Is anyone is able to help me at this point?

Answer 1

Hint: Expand $(x+2)^p-2^p$ and remember that, if $p$ is prime, $p$ is a divisor of all $\dbinom pk$ for $0<k<p$.