How was this factoring of $1-2x-x^2$ achieved? If I factor $1-2x-x^2$ using the quadratic formula I get
$$x=\frac{2\pm \sqrt{4-4(-1)(1)}}{2(-1)}$$
$$x=\frac{2\pm \sqrt{8}}{-2}$$
$$x=-1 \pm \sqrt{2}$$
Let $\alpha = -1 +\sqrt{2}$ and $\beta=-1-\sqrt{2}$. 
So $1-2x-x^2=-(x-\alpha)(x-\beta)$.
In the image below, where did $1-\alpha x$ and $1-\beta x$ come from?



That's an awful lot of work for such a simple factoring problem.
Take out a negative: $$-[x^2+2x-1]$$
Then complete the square inside: $$-[(x+1)^2-2]$$
Use the difference of two squares (Of course $2$ is a square...) $$-(x+1+\sqrt{2})(x+1-\sqrt{2})$$
Is this sufficient?
