a) (x - 1) . (x5 + x4 + x3 + x2 + x + 1) = (x . x5 + x . x4 + x . x3 + x . x2 + x . x + x . 1) - (1 . x5 + 1 . x4 + 1 . x3 + 1 . x2 + 1 . x + 1 . 1)
= (x6 + x5 + x4 + x3 + x2 + x ) - (x5 + x4 + x3 + x2 + x + 1)
= x6 + x5 + x4 + x3 + x2 + x - x5 - x4 - x3 - x2 - x - 1
= x6 + (x5 - x5) + (x4 - x4) + (x3 - x3) + (x2 - x2) + (x - x) - 1
= x6 - 1
b) (x + 1) . (x6 - x5 + x4 - x3 + x2 - x + 1) = (x . x6 - x . x5 + x . x4 - x . x3 + x . x2 - x . x + x . 1) + (1 . x6 - 1 . x5 + 1 . x4 - 1 . x3 + 1 . x2 - 1 . x + 1 . 1)
= (x7 - x6 + x5 - x4 + x3 - x2 + x ) + (x6 - x5 + x4 - x3 + x2 - x + 1)
= x7 - x6 + x5 - x4 + x3 - x2 + x + x6 - x5 + x4 - x3 + x2 - x + 1
= x7+(-x6 + x6) + (x5 - x5) + (-x4 + x4) + (x3 - x3) + (-x2 + x2) + (x - x) + 1
= x7 + 1
