This expression can be factored as a difference of two squares because 81 is a square number. Rewrite 81 as `x^4` as `(x^2)^2`, `9^2 = (3^2)^2` and `y^4` as `(y^2)^2`,

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

Combine `(3^2)^2` and `(y^2)^2`,

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

Applying the difference of squares formula,

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

Applying the difference of squares formula on the second parenthesis,

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

Simplify by rewriting (3y)^2 as 9y^2,

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

Thus the expression `x^4 - 81y^4` is completely factored to `(x^2 + 9y^2)(x + 3y)(x - 3y)`

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

Combine `(3^2)^2` and `(y^2)^2`,

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

Applying the difference of squares formula,

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

Applying the difference of squares formula on the second parenthesis,

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

Simplify by rewriting (3y)^2 as 9y^2,

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

Thus the expression `x^4 - 81y^4` is completely factored to `(x^2 + 9y^2)(x + 3y)(x - 3y)`

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