Simplify each of the following expressions:$ (x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y $

Given:

\( (x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y \)

To do:

We have to simplify \( (x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y \).

Solution:

We know that,

$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$

Therefore,

$(x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y=x^{2}+y^{2}+4 z^{2}+2 x y-4 y z-4 z x-x^{2}-y^{2}-3 z^{2}+4 x y$

$=z^{2}+6 x y-4 y z-4 z x$

Hence, $(x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y=z^{2}+6 x y-4 y z-4 z x$.