1. **Problem:** Factorize completely the expression $x^4 - 81$. 2. **Formula and rules:** Recognize that $x^4 - 81$ is a difference of squares because $x^4 = (x^2)^2$ and $81 = 9^2$. The difference of squares formula is: $$a^2 - b^2 = (a - b)(a + b)$$ 3. **Apply the formula:** $$x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9)$$ 4. **Further factorization:** Notice that $x^2 - 9$ is also a difference of squares: $$x^2 - 9 = (x - 3)(x + 3)$$ 5. **Final factorization:** $$x^4 - 81 = (x - 3)(x + 3)(x^2 + 9)$$ 6. **Explanation:** We first used the difference of squares formula to break down the original expression into two factors. Then, we factored the $x^2 - 9$ term further because it is also a difference of squares. The term $x^2 + 9$ cannot be factored further over the real numbers. 7. **Answer choice:** The factorization matches option B: $(x^2 - 9)(x^2 + 9)$. **Final answer:** $(x^2 - 9)(x^2 + 9)$