a: \(x^4-7x^3+14x^2-7x+1\)
\(=x^4-4x^3+x^2-3x^3+12x^2-3x+x^2-4x+1\)
\(=x^2\left(x^2-4x+1\right)-3x\left(x_{}^2-4x+1\right)+\left(x^2-4x+1\right)=\left(x^2-4x+1\right)\left(x^2-3x+1\right)\)
b: \(x^4+6x^3+7x^2-6x+1\)
\(=x^4+3x^3-x^2+3x^3+9x^2-3x-x^2-3x+1\)
\(=x^2\left(x^2+3x-1\right)+3x\left(x^2+3x-1\right)-\left(x^2+3x-1\right)=\left(x^2+3x-1\right)^2\)
c: \(\left(x^2+2x+8\right)^2+3x\left(x^2+2x+8\right)+2x^2\)
\(=\left(x^2+2x+8\right)^2+x\left(x^2+2x+8\right)+2x\left(x^2+2x+8\right)+2x^2\)
\(=\left(x^2+2x+8\right)\left(x^2+2x+8+x\right)+2x\left(x^2+2x+8+x\right)\)
\(=\left(x^2+3x+8\right)\left(x^2+4x+8\right)\)
d: \(\left(x^2-x+1\right)^2-5x\left(x^2-x+1\right)+4x^2\)
\(=\left(x^2-x+1\right)^2-x\left(x^2-x+1\right)-4x\left(x^2-x+1\right)+4x^2\)
\(=\left(x^2-x+1\right)\left(x^2-x+1-x\right)-4x\left(x^2-x+1-x\right)\)
\(=\left(x^2-x+1-4x\right)\left(x^2-2x+1\right)=\left(x-1\right)^2\cdot\left(x^2-5x+1\right)\)
