1. The problem is to simplify the expression $81 - p^4$. 2. Recognize that $81$ is a perfect square since $81 = 9^2$. 3. The expression $81 - p^4$ can be rewritten as $9^2 - (p^2)^2$. 4. This is a difference of squares, which follows the formula $a^2 - b^2 = (a - b)(a + b)$. 5. Applying the formula, we get: $$81 - p^4 = (9 - p^2)(9 + p^2)$$ 6. The factor $9 - p^2$ is also a difference of squares since $9 = 3^2$. 7. Factor $9 - p^2$ further: $$9 - p^2 = (3 - p)(3 + p)$$ 8. Therefore, the fully factored form is: $$(3 - p)(3 + p)(9 + p^2)$$ Final answer: $81 - p^4 = (3 - p)(3 + p)(9 + p^2)$