a: \(\left(x+1\right)^3=x^3+3\cdot x^2\cdot1+3\cdot x\cdot1^2+1^3\)
\(=x^3+3x^2+3x+1\)
b: \(\left(x-2\right)^3=x^3-3\cdot x^2\cdot2+3\cdot x\cdot2^2-2^3=x^3-6x^2+12x-8\)
c: \(\left(2-x\right)^3=2^3-3\cdot2^2\cdot x+3\cdot2\cdot x^2-x^3=8-12x+6x^2-x^3\)
d: \(\left(x^2-y\right)^3=\left(x^2\right)^3-3\cdot\left(x^2\right)^2\cdot y+3\cdot x^2\cdot y^2-y^3\)
\(=x^6-3x^4y+3x^2y^2-y^3\)
e: \(\left(x-1\right)^3-x\left(x-2\right)^2+x-1\)
\(=x^3-3x^2+3x-1-x\left(x^2-4x+4\right)+x-1\)
\(=x^3-3x^2+4x-2-x^3+4x^2-4x=x^2-2\)
f: \(\left(x+2\right)^3=x^3+3\cdot x^2\cdot2+3\cdot x\cdot2^2+2^3=x^3+6x^2+12x+8\)
\(\left(x+2\right)^3-x^2\left(x+6\right)-8\)
\(=x^3+6x^2+12x+8-x^3-6x^2-8\)
=12x
