f(x) = (x^2 + x + 1)Q(x) + 2x-1

{f(x)}^2 = {(x^2 + x + 1)Q(x) + 2x-1}^2
≡(2x-1)^2   (mod:x^2 + x + 1)
=4x^2-4x+1
=4(x^2+x+1)-8x-3
≡-8x-3   (mod:x^2 + x + 1)