a) Ta có: \(A=\left(x-1\right)^3-\left(x+1\right)^3\)
\(=\left[\left(x-1\right)-\left(x+1\right)\right]\cdot\left[\left(x-1\right)^2+\left(x-1\right)\left(x+1\right)+\left(x+1\right)^2\right]\)
\(=\left(x-1-x-1\right)\cdot\left(x^2-2x+1+x^2-1+x^2+2x+1\right)\)
\(=-2\cdot\left(3x^2+1\right)\)
\(=-6x^2-2\)
b) Ta có: \(B=\left(x+y\right)^3+\left(x-y\right)^3\)
\(=\left[\left(x+y\right)+\left(x-y\right)\right]\cdot\left[\left(x+y\right)^2-\left(x+y\right)\left(x-y\right)+\left(x-y\right)^2\right]\)
\(=\left(x+y+x-y\right)\left(x^2+2xy+y^2-x^2+y^2+x^2-2xy+y^2\right)\)
\(=2x\cdot\left(x^2+3y^2\right)\)
\(=2x^3+6xy^2\)
c) Ta có: \(C=\left(x-y\right)^3+3xy\left(x-y\right)\)
\(=\left(x-y\right)\left(x^2-2xy+y^2\right)+3xy\left(x-y\right)\)
\(=\left(x-y\right)\left(x^2-2xy+y^2+3xy\right)\)
\(=\left(x-y\right)\left(x^2+xy+y^2\right)\)
\(=x^3-y^3\)
d) Ta có: \(D=\left(x+1\right)^3-\left(x-3\right)^3-2\left(x^2+15\right)\left(x-3\right)\)
\(=x^3+3x^2+3x+1-\left(x^3-9x^2+27x-27\right)-2\left(x^3-3x^2+15x-45\right)\)
\(=x^3+3x^2+3x+1-x^3+9x^2-27x+27-2x^3+6x^2-30x+90\)
\(=-2x^3+18x^2-54x+118\)
