Why is $a^4+b^4$ factorable in two different ways, and why are the solutions not the same? I am working on factoring $a^4+b^4$ and I have found two different solutions to this.
First, I have factored it to $$a^4+b^4=(a+b)[a^3-a^2b+ab^2+b^3]-2ab^3$$
But then I also found that the equation is factorable to $$a^4+b^4=(a^2+b^2-\sqrt{2}ab)(a^2+b^2+\sqrt{2}ab)$$
First of all, why are there two ways of factoring this equation, and why are the solutions not the same?
Additionally, is it possible to convert either solution so it is the same as the other?

A factorization is always a pure product, e.g.,
$$a^3+b^3=(a+b)(a^2-ab+b^2)$$
or
$$a^4-b^4=(a-b)(a+b)(a^2+b^2)$$
So the problem is that $(a+b)(a^3-a^2b+ab^2+b^3)-2ab^3$, is not a pure product.  Instead, it's the sum of the product $(a+b)(a^3-a^2b+ab^2+b^3)$ and $-2ab^3$.
I.e., what tilper just said!
