a ) \(\left(x+y\right)^2-2\left(x+y\right)+1=\left(x+y-1\right)^2\)
b ) \(\left(a+b+c\right)^3-a^3-b^3-c^3\)
\(=\left(a+b\right)^3+3\left(a+b\right)^2c+3\left(a+b\right)c^2+c^3-a^3-b^3-c^3\)
\(=a^3+3a^2b+3b^2a+b^3+3\left(a^2+2ab+b^2\right)c+3ac^2+3bc^2+c^3-a^3-b^3-c^3\)\(=3a^2b+3b^2a+3a^2c+6abc+3b^2c+3ac^2+3bc^2\)
\(=3\left(a^2b+b^2a+a^2c+abc+abc+b^2c+ac^2+bc^2\right)\)
\(=3\left[\left(a^2b+b^2a\right)+\left(a^2c+abc\right)+\left(abc+b^2c\right)+\left(ac^2+bc^2\right)\right]\)
\(=3\left[ab\left(a+b\right)+ac\left(a+b\right)+bc\left(a+b\right)+c^2\left(a+b\right)\right]\)
\(=3\left(a+b\right)\left(ab+ac+bc+c^2\right)\)
\(=3\left(a+b\right)\left[a\left(b+c\right)+c\left(b+c\right)\right]\)
\(=3\left(a+b\right)\left(b+c\right)\left(a+c\right)\)
c ) \(a^3+b^3+c^3-3abc\)
\(=\left(a^3+3a^2b+3b^2a+b^3\right)+c^3-3abc-3b^2a-3ab^2\)
\(=\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left[a^2+2ab+b^2-ac-bc+c^2\right]-3ab\left(a+b+c\right)\)\(=\left(a+b+c\right)\left(a^2+2ab+b^2+ac+bc+c^2-3ab\right)\)
\(=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-ac-bc\right)\)
