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.