1. **State the problem:** Factor the expression $x^4 - 81$. 2. **Recognize the form:** This is a difference of squares since $81 = 9^2$ and $x^4 = (x^2)^2$. 3. **Apply the difference of squares formula:** $$a^2 - b^2 = (a - b)(a + b)$$ Here, $a = x^2$ and $b = 9$. 4. **Factor the expression:** $$x^4 - 81 = (x^2 - 9)(x^2 + 9)$$ 5. **Further factor $x^2 - 9$:** Again, this is a difference of squares: $$x^2 - 9 = (x - 3)(x + 3)$$ 6. **Final factorization:** $$x^4 - 81 = (x - 3)(x + 3)(x^2 + 9)$$ 7. **Explanation:** We used the difference of squares formula twice to break down the original expression into simpler factors. The term $x^2 + 9$ cannot be factored further over the real numbers because it is a sum of squares. **Answer:** $$\boxed{(x - 3)(x + 3)(x^2 + 9)}$$