Factor x12 - x4 - x3 + 1

Factor only the first two terms by taking out x4

     x4(x8 - 1) - x3 + 1

Factor only the last two terms by taking out -1

     x4(x8 - 1) - 1(x3 - 1)

Erase the 1

     x4(x8 - 1) - (x3 - 1)

Factor (x8 - 1) as the difference of squares (x4 + 1)(x4 - 1)

     x5(x4 + 1)(x4 - 1) - (x3 - 1)

Factor the (x4 - 1) as the difference of squares (x2 - 1)(x2 - 1)

     x4(x4 + 1)(x2 - 1)(x2 + 1) - (x3 - 1)

Factor the (x2 - 1) as the difference of squares (x - 1)(x + 1)

     x4(x4 + 1)(x - 1)(x + 1)(x2 + 1) - (x3 - 1) 

Factor the (x3 - 1) as the sum of cubes (x - 1)(x2 + x + 1)

     x4(x4 + 1)(x - 1)(x + 1)(x2 + 1) - (x - 1)(x2 + x + 1)

Factor out (x - 1) using brackets:

   (x - 1)[x4(x4 + 1)(x + 1)(x2 + 1) - (x2 + x + 1)]

Remove all the parentheses inside the brackets:

   (x - 1)[(x8 + x4)(x3 + x + x2 + 1) - x2 - x - 1]

   (x - 1)[x11 + x9 + x10 + x8 + x7 + x5 + x6 + x4 - x2 - x - 1]

Rearrange the terms in the brackets in descending order and change
the brackets to parentheses:

   (x - 1)(x11 + x10 + x9 + x8 + x7 + x6 + x5 + x4 - x2 - x - 1)

That's it!  It won't factor any further than that.

Edwin