If a polynomial with integer coefficients cannot be factored into two polynomials of lower degree with rational coefficients, then certainly, you can't do it over Z either. So what am I missing here?
In $\mathbb Q$, which is a field, all non-zero elements are units, so there's no such thing as a common constant factor; factoring a polynomial over $\mathbb Q$ means factoring it into two polynomials of lower degree. By contrast, over $\mathbb Z$ you can factor a polynomial into a constant factor common to all coefficients and a polynomial of the same degree. Of course the same "factorization" also holds over $\mathbb Q$, but over $\mathbb Q$ the constant factor is a unit, so this doesn't count as factoring the polynomial.