MHB Can You Successfully Factorize x^2+y^2+z^2-2xy-2yz-2zx?

[anemone]

Factorize $x^2+y^2+z^2-2xy-2yz-2zx$.

 

[lfdahl]

My attempt:

Spoiler

In order to perform the factorization, I must require: $y,z \ge 0$.

\[x^2+y^2+z^2-2xy-2zx-2yz \\\\= (x-y-z)^2-4yz \\\\=(x-y-z+2\sqrt{yz})(x-y-z-2\sqrt{yz}) \\\\=(x-((\sqrt{y})^2+(\sqrt{z})^2-2\sqrt{yz}))(x-((\sqrt{y})^2+(\sqrt{z})^2+2\sqrt{yz})) \\\\=(x-(\sqrt{y}-\sqrt{z})^2)(x-(\sqrt{y}+\sqrt{z})^2)\]

 

[anemone]

My attempt:

Spoiler

In order to perform the factorization, I must require: $y,z \ge 0$.

\[x^2+y^2+z^2-2xy-2zx-2yz \\\\= (x-y-z)^2-4yz \\\\=(x-y-z+2\sqrt{yz})(x-y-z-2\sqrt{yz}) \\\\=(x-((\sqrt{y})^2+(\sqrt{z})^2-2\sqrt{yz}))(x-((\sqrt{y})^2+(\sqrt{z})^2+2\sqrt{yz})) \\\\=(x-(\sqrt{y}-\sqrt{z})^2)(x-(\sqrt{y}+\sqrt{z})^2)\]

Very Well done, lfdahl! (Cool)