Factor `x^4 - y^4`

This expression is factored using the difference of squares formula. Rewrite `x^4` as (x^2)^2` and `y^4` as (y^2)^2`,

`x^4 - y^4 = (x^2)^2 - (y^2)^2`

Now this expression is in the form of a difference of two squares, the first 'square' is `(x^2)^2` and the second square is `(y^2)^2`. Apply the difference of squares formula `a^2 - b^2 = (a + b(a - b)`,

`(x^2)^2 - (y^2)^2 = (x^2 + y^2)(x^2 - y^2)`

The first parenthesis above can not be factored since it is not a difference of two squares but a sum of two squares, and you don't have a factoring formula for a sum of two squares. The second parenthesis is a difference of two squares since there is a minus sign in between and two squared terms are subtracted, so applying the difference of squares formula on it,

`(x^2 + y^2)(x^2 - y^2) = (x^2 + y^2)(x + y)(x - y)`

Thus you have completely factored the expression `x^2 - y^4` into `(x^2 + y^2)(x + y)(x - y)`

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