Algebra: factoring by grouping?

x^3 + x^2 - x - 1

Take x^2 as a factor from the first two terms and -1 as from the last two terms:

x^2 (x+1) - 1 (x+1)

(x^2 - 1) (x+1) ------------------- (1)

x^2 - 1 above can be written as, (x^2 - 1^2), which can be written as (x-1) (x+1). Plugging it back in result (1)

(x+1)^2 (x-1)

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The next one is the same.

Factor a^2 from the first two terms and -8 from the last two terms.

a^2 (a - 1) - 8 (a - 1)

(a^2 - 8) (a - 1)

x^2(x+1)-(x+1)=(x+1)(x+1)(x-1)

1)
x^3 + x^2 - x - 1
= (x^3 + x^2) - (x + 1)
= x^2(x + 1) - 1(x + 1)
= (x + 1)(x^2 - 1)
= (x + 1)(x^2 - 1^2)
= (x + 1)(x + 1)(x - 1)
= (x + 1)^2(x - 1)

2)
a^3 - a^2 - 8a + 8
= (a^3 - a^2) - (8a - 8)
= a^2(a - 1) - 8(a - 1)
= (a - 1)(a^2 - 8)