1) \(a^4-3a^3-6a^3+18a^2-18a^2+54a+27a-81\)
\(=a^3\left(a-3\right)-6a^2\left(a-3\right)-18a\left(a-3\right)+27\left(a-3\right)\)
\(=\left(a-3\right)\left(a^3-6a^2-18a+27\right)\)
\(=\left(a-3\right)\left(a^3+3a^2-9a^2-27a+9a+27\right)\)
\(=\left(a-3\right)\left[a^2\left(a+3\right)-9a\left(a+3\right)+9\left(a+3\right)\right]\)
\(=\left(a-3\right)\left(a+3\right)\left(a^2-9a+9\right)\)
2) Ta có:
\(\left(x+y+z\right)^3-x^3-y^3-z^3\)
\(=\left[\left(x+y+z\right)-x\right]\left[\left(x+y+z\right)^2+x\left(x+y+z\right)+x^2\right]-\left(y+z\right)\left(y^2-yz+z^2\right)\)
\(=\left(y+z\right)\left(x^2+y^2+z^2+2xy+2xz+2yz+x^2+xy+xz+x^2\right)-\left(y+z\right)\left(y^2-yz+z^2\right)\)
\(=\left(y+z\right)\left(3x^2+3xy+3xz+2yz+y^2+z^2\right)-\left(y+z\right)\left(y^2-yz+z^2\right)\)
\(=\left(y+z\right)\left(3x^2+3xy+3xz+2yz+y^2+z^2-y^2+yz-z^2\right)\)
\(=\left(y+z\right)\left(3x^2+3xy+3xz+3yz\right)\)
\(=3\left(y+z\right)\left(x^2+xy+xz+yz\right)\)
\(=3\left(y+z\right)\left[x\left(x+y\right)+z\left(x+y\right)\right]\)
\(=3\left(y+z\right)\left(x+y\right)\left(x+z\right)\)