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### 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)