I was pondering on the idea of a difference of two square, and if one can apply the same procedure for a sum of two square: $x^2 - y^2 = (x-y)(x+y)$. I thought I could also make the form $x^2 + y^2$ in the form of a difference of two square but it turns out I couldn't. This is the procedure I followed:
$$x^2 + y^2 = x^2 - (-y)^2 = (x - y)(x + y) = x^2 - y^2$$ which is not possible. Why did this happen?
$$ -(-y)^2 = -y^2$$
In general we do not have $y^2 = -(-y)^2$
We do have $$x^2 + y^2 = x^2 -(iy)^2=(x+iy)(x-iy)$$