Factor `x^4 - 81`

This expression can be factored as a difference of two squares, because 81 is a square number and 81 can be written as `3^4`,

`x^4 - 3^4`

Rewrite `x^4` as `(x^2)^2` and `3^4` as `(3^2)^2`,

`(x^2)^2 - (3^2)^2`

Apply the difference of two squares formula `a^2 - b^2 = (a + b)(a - b)`

`(x^2 + 3^2)(x^2 - 3^2)`

Again apply the difference of squares formula,

`(x^2 + 3^2)(x + 3)(x - 3)`

Simplify by writing `3^2` as 9,

`(x^2 + 9)(x + 3)(x - 3)`

Thus we have completely factored the expression `x^4 - 81` into `(x^2 + 9)(x + 3)(x - 3)`