a: \(x^3+6x^2+11x+6\)
\(=x^3+x^2+5x^2+5x+6x+6\)
\(=x^2\left(x+1\right)+5x\left(x+1\right)+6\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2+5x+6\right)=\left(x+1\right)\left(x+2\right)\left(x+3\right)\)
b: \(x^3-8x^2+x+42\)
\(=x^3+2x^2-10x^2-20x+21x+42\)
\(=x^2\left(x+2\right)-10x\left(x+2\right)+21\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-10x+21\right)=\left(x+2\right)\left(x-3\right)\left(x-7\right)\)
c: \(x^3-4x^2+3\)
\(=x^3-x^2-3x^2+3\)
\(=x^2\left(x-1\right)-3\left(x-1\right)\left(x+1\right)\)
\(=\left(x-1\right)\left(x^2-3x-3\right)\)
d: \(2x^3+3x^2+2x-2\)
\(=2x^3-x^2+4x^2-2x+4x-2\)
\(=x^2\left(2x-1\right)+2x\left(2x-1\right)+2\left(2x-1\right)=\left(2x-1\right)\left(x^2+2x+2\right)\)
e: \(6x^3-11x^2-x-2\)
\(=6x^3-12x^2+x^2-2x+x-2\)
\(=6x^2\left(x-2\right)+x\left(x-2\right)+\left(x-2\right)=\left(x-2\right)\left(6x^2+x+1\right)\)
f: \(4x^3+5x^2+5x+1\)
\(=4x^3+x^2+4x^2+x+4x+1\)
\(=x^2\left(4x+1\right)+x\left(4x+1\right)+\left(4x+1\right)=\left(4x+1\right)\left(x^2+x+1\right)\)
