Solution :

Suppose that P(x) is a product of two nonconstant polynomials. Then these polynomials are monic and both have degree 2 since the coefficient of P(x) at x ⁴  is 1 and P(x) > 0 has no real roots:

At x = a and x = b  P(x) = 1, therefore  at x = a and x = b. Thus,  has two different roots a and b, and therefore is identically zero: p = r and q = s.

Then we have . Since the only two squares that differ by 1 are 0 and 1, , a contradiction.