1. \(x^3+2x^2+2x+1\)
\(=\left(x^3+1\right)+\left(2x^2+2x\right)\)
\(=\left(x+1\right)\left(x^2+x+1\right)+2x\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2+3x+1\right)\)
2. \(x^3-4x^2+12x-27\)
\(=x^3-3x^2-x^2+3x+9x-27\)
\(=x^2\left(x-3\right)-x\left(x-3\right)+9\left(x-3\right)\)
\(=\left(x^2-x+9\right)\left(x-3\right)\)
3. \(x^4-2x^3+2x-1\)
\(=\left(x^4-1\right)-2x\left(x^2-1\right)\)
\(=\left(x^2-1\right)\left(x^2+1\right)-2x\left(x^2-1\right)\)
\(=\left(x^2-1\right)\left(x^2-2x+1\right)\)
\(=\left(x-1\right)\left(x+1\right)\left(x-1\right)^2\)
\(=\left(x-1\right)^3\left(x+1\right)\)
d) \(x^4+2x^3+2x^2+2x+1\)
\(=x^4+x^2+2x^3+2x+x^2+1\)
\(=x^2\left(x^2+1\right)+2x\left(x^2+1\right)+\left(x^2+1\right)\)
\(=\left(x^2+2x+1\right)\left(x^2+1\right)\)
\(=\left(x+1\right)^2\left(x^2+1\right)\)