Why $3X^3+12X^2+30$ is irreducible on $\mathbb Q$ and not in $\mathbb Z$?

Question

I don't understand why $3X^3+12X^2+30$ is irreducible on $\mathbb Q$ and not in $\mathbb Z$ ? Indeed, don't we have $$3X^3+12X^2+30=3(X^3+4X^2+10)$$ is a factorization in $\mathbb Q$ ? If it's a factorization in $\mathbb Z$, why it is not consider as a factorization in $\mathbb Q$ ?

Answer 1

Because $3$ is invertible in $\mathbb Q$ and not in $\mathbb Z$.

Answer 2

$X^3+4X+10$ is irreducible over $\mathbb Q$ which is seen, for example, by the Einsenstein criterion of irreducibility, and stay irreducible over $\mathbb Q$ if it is multiplied for an arbitrary unit (i.e. invertible element) of $\mathbb Q$, in particular for your $3$. However the constant polynomial $p(X)=3$ (which is invertible in $\mathbb Q[X]$, having as inverse $q(X)=\frac13$) is not invertible in $\mathbb Z[X]$ because the only units of $\mathbb Z$ are $1$ and $-1$.