Proof by Contradiction: Polynomial cannot be written as the product of two quadratics
I've been working through Chapter 1 of D. L. Johnson's "Elements of Logic via Numbers and Sets" and have become stumped on the following problem, amongst others.
"Prove that the polynomial cannot be written as a product of two quadratic polynomials with integer coefficients."
The preceding section (and indeed the other questions in this set) was concerned with 'proof by contradiction', so I assume that I am intended to solve this via this method.
Therefore, for contradiction, assume that can be written as previously described, i.e.
where and are integers. Thus,
and by comparing coefficients we see that:
I suppose for the purposes of the proof then, I should aim to show that there are no integer solutions to these set of simultaneous equations - but I have no idea about how to approach this. Any suggestions would be much appreciated.