Algebra

(X^8)-1 =((x^4)^2)-1^2=[((x^2)^2) -1^2][(x^4)+1]
=[(x^2)-1^2][(x^2)+1][(x^4)+1]
=[x-1][x+1][(x^2)+1][(x^4)+1]

(x^8) -1

since 8 can be divide by two
8 / 2 = 4

definition of sum and difference
a^2 - b^2 = (a - b)(a + b)

we get,
((x^4) - 1)((x^4) + 1)


since 4 can be divide by two
4 / 2 = 2

((x^4) + 1)((x^2) + 1)((x^2) - 1)


since 2 can be divide by two
2 / 2 = 1

((x^4) + 1)((x^2) + 1)(x - 1)(x + 1)

(4x+1)(4x-1)