a: \(\left(a+b-c\right)^2+\left(a-b+c\right)^2-2\left(b-c\right)^2\)
\(=a^2+2a\left(b-c\right)+\left(b-c\right)^2+a^2-2a\left(b-c\right)+\left(b-c\right)^2-2\left(b-c\right)^2\)
\(=2a^2\)
b: \(\left(a+b+c\right)^2+\left(a-b-c\right)^2+\left(b-c-a\right)^2+\left(c-a-b\right)^2\)
\(=\left(a+b+c\right)^2+\left(a-b-c\right)^2+\left(a-b+c\right)^2+\left(a+b-c\right)^2\)
\(=a^2+2a\left(b+c\right)+\left(b+c\right)^2+a^2-2a\left(b+c\right)+\left(b+c\right)^2+a^2-2a\left(b-c\right)+\left(b-c\right)^2+a^2+2a\left(b-c\right)+\left(b-c\right)^2\)
\(=2a^2+2\left(b+c\right)^2+2a^2+2\left(b-c\right)^2\)
\(=4a^2+2\left(b^2+2bc+c^2+b^2-2bc+c^2\right)\)
\(=4a^2+2\left(2b^2+2c^2\right)=4a^2+4b^2+4c^2\)
c: \(\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2\)
\(=\left(a+b\right)^2+2\left(a+b\right)\left(c+d\right)+\left(c+d\right)^2+\left(a+b\right)^2-2\left(a+b\right)\left(c+d\right)+\left(c+d\right)^2\)
\(=2\left(a+b\right)^2+2\left(c+d\right)^2\)
\(\left(a+c-b-d\right)^2+\left(a+d-b-c\right)^2\)
\(=\left(a-b+c-d\right)^2+\left(a-b-c+d\right)^2\)
\(=\left(a-b\right)^2+2\left(a-b\right)\left(c-d\right)+\left(c-d\right)^2+\left(a-b\right)^2-2\left(a-b\right)\left(c-d\right)+\left(c-d\right)^2\)
\(=2\left(a-b\right)^2+2\left(c-d\right)^2\)
\(\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2+\left(a+c-b-d\right)^2+\left(a+d-b-c\right)^2\)
\(=2\left(a+b\right)^2+2\left(c+d\right)^2+2\left(a-b\right)^2+2\left(c-d\right)^2\)
\(=2\left(a^2+2ab+b^2+a^2-2ab+b^2\right)+2\left(c^2+2cd+d^2+c^2-2cd+d^2\right)\)
\(=2\left(2a^2+2b^2\right)+2\left(2c^2+2d^2\right)=4a^2+4b^2+4c^2+4d^2\)
