Simplify the Following Expression
\[ (x+y-2z)^2-x^2-y^2-3z^2+4xy \]
Solution:
Using identity:
\[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \]
\[ (x+y-2z)^2 = x^2+y^2+(2z)^2+2xy-2(y)(2z)-2(x)(2z) \]
\[ = x^2+y^2+4z^2+2xy-4yz-4xz \]
\[ (x+y-2z)^2-x^2-y^2-3z^2+4xy \]
\[ = x^2+y^2+4z^2+2xy-4yz-4xz -x^2-y^2-3z^2+4xy \]
\[ = z^2+6xy-4yz-4xz \]