Factoring binary polynomials

Question

I need to factor two binary polynomials and present each as a product of powers of irreducible polynomials.

a) x⁴ + 1

I have figured it out this far: x⁴ = (x²)² and 1 = 1²

So I have something in the form a²-b² = (a+b)(a-b) So first factorization: (x²+1)(x²-1)
Then, x²-1 is also in a²-b² form: Second factorization: (x²+1)(x+1)(x-1)

Is this the final answer? Is it in the form of a product of powers of irreducible polynomials? Thanks!

Answer 1

it is $x^4+1=x^4+2x^2+1-2x^2=(x^2+1)^2-2x^2=(x^2+1-\sqrt{2}x)(x^2+1+\sqrt{2}x)$