x^6-1 = x^6-1
Apply (1) to the left side and (2) to the right side:
(x^3-1)(x^3+1) = (x^2-1)(x^4+x^2+1)
Next, apply (2) and (3) to the respective factors on the left side and apply (1) to (x^2-1) on the right side
(x-1)(x^2+x+1)(x+1)(x^2-x+1) = (x-1)(x+1)(x^4+x^2+1)
Since (x-1)(x+1) appears on each side, the factorization of x^4+x^2+1 is (x^2+x+1)(x^2-x+1)
Check the comments for more ways to solve the problem, some of which include other equally basic factoring identites and methods.