Gauss' Lemma: If the primitive polynomial $f(x)$ can be factored as the product of two polynomials having rational coefficients, it can be factored as the product of two polynomials having integer coefficients.
Corollary: If an integer monic polynomial factors as the product of two non-constant polynomials having rational coefficients then it factors as the product of two integer monic polynomials.
I have proved the Gauss' lemma using the fact that product of two primitive polynomials is also primitive.
I guess that corollary follows easily from the Gauss' lemma since integer monic polynomial is primitive and then comparing the leading coefficients leads to $ab=1$ which is equivalent to $a=b=1$ or $a=b=-1$. In both cases we get the product of two integer monic polynomial.
Let me ask you the following question: Why the condition of corollary contains non-constant polynomials? What if one of them is constant?
EDIT: I have checked that if $f(x)$ is an integer monic polynomial and it factors as the product of two polynomials from $\mathbb{Q}[x]$ where one of them is constant then statement of corollary is also true.