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

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?

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