a) `4x^4 +4x^3 +5x^3 +2x+1`
`=4x^4 +4x^3 +x^2 +4x^2 +2x+1`
`=x.(2x+1)^2 +(2x+1)^2`
`=(x+1)(2x+1)^2`
b) `x^4 -7x^3 +14x^2 -7x+1`
`=x^4 -3x^3 +1x^2 -4x^3 +12x^2 -4x+x^2 -3x+1`
`=(x^2 -3x+1).x^2 -4(x^3 -3x^2 +x)+x^2 -3x+1`
`=(x^2 -3x+1).x^2 -4x(x^2 -3x+1)+1(x^2 -3x+1)`
`=(x^2 -4x+1)(x^2 -3x+1)`
c) `x^4 -8x+63`
`=x^4 +4x^3 +9x^2 -4x^3 -16x^2 -36x +7x^2 +28x +63`
`=(x^2 +4x+9).x^2 -4x(x^2 +4x+9)+7(x^2 +4x+9)`
`=(x^2 -4x+7)(x^2 +4x+9)`
d) `(x+1)^4 +(x^2 +x+1)^2`
`=(x+1)^4 +x^2 (x+1)^2 +2x(x+1)+1`
`=(x+1)^2 (2x^2 +2x+1)+(2x^2 +2x+1)`
`=(2x^2 +2x+1)(x^2 +2x+2)`
