a) x5 + x + 1
= x5 - x4 + x4 - x3 + x3 - x2 + x2 + x + 1
= ( x5 + x4 + x3) - ( x4 + x3 + x2) + (x2 + x +1)
= x3( x2 + x + 1) - x2( x2 + x + 1) + (x2 + x +1)
= ( x2 + x +1).( x3 - x2 + 1)
b) ( x2 + x)2 -2(x2 + x) -15
=( x2 + x)2 -2(x2 + x).1 + 1- 16
=( x2 + x - 1)2 - 42
=( x2 + x - 1 - 4).( x2 + x - 1+ 4)
=(x2 + x - 5).( x2 + x + 3)
c) x4 + 5x3 + 10x - 4
= (x2)2 - 22 + 5x.( x2 + 2)
=( x2 -2).(x2 + 2) + 5x.( x2 + 2)
= ( x2 + 2).(x2 -2 + 5x)
d) x8 + x7 + 1
= x8 + x7 + x6 - x6 + 1
= x6 ( x2 + x + 1) - ( x6 - 1)
= x6( x2 + x + 1) - ( x3 - 1).(x3 + 1)
= x6( x2 + x + 1) - ( x- 1).( x2 + x + 1).(x3 + 1)
= ( x2 + x + 1).[ x6 -( x- 1).(x3 + 1)]
= ( x2 + x + 1).( x6 - x4 + x3 - x +1)
