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.