How do you factor x4 plus y4?

You can't.

x4 + y4 is irreducible.

It's (x+yi^1.5)(x+yi^0.5)(x-yi^1.5)(x-yi^0.5). You use the equation x^2-y^2=(x+y)(x-y) and x^2+y^2=(x+yi)(x-yi).

*Edit This IS reducible.

X^4+Y^4=X^4+Y^4+2X^2Y^2 -2X^2Y^2 I add and minus 2X^2Y^2 so the value stays the same.

X^4+Y^4+2X^2Y^2=(X^2+Y^2)^2

So it equals to (X^2+Y^2)^2-2X^2Y^2.

Then, we factor using difference of squares.

(X^2+Y^2)^2-2X^2Y^2=(X^2+Y^2+sqrt2 XY)(X^2+Y^2-sqrt2 XY)

*Note the XY is not included in the sqrt sign