The proof begins like this.
So we suppose $f(x)= a_0+a_1x+...+a_nx^n$ is reducible and thus
$f(x) = g(x)h(x) = (b_0+b_1x+...+b_rx^r)(c_0+c_1x+...+c_sx^s$).
Then $a_0=b_0c_0$. By the definition of Eisenstein's criterion $p\mid a_0$ and so $p\mid b_0$ or $p\mid c_0$. Lets say $p\mid b_0$. "Since $p^2$ does not divide $a_0$ (by definition of Eisentein's criterion), we see that $c_0$ is not divisible by $p$."
I don't understand that last statement. How does $c_0$ not being divisible by $p$ follow from the fact that $p^2$ does not divide $a$?