Rewrite $81-x^{4}$ as $9^{2}-\left(-x^{2}\right)^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(9+x^{2}\right)\left(9-x^{2}\right)$$
Reorder the terms.
$$\left(x^{2}+9\right)\left(-x^{2}+9\right)$$
Consider $-x^{2}+9$. Rewrite $-x^{2}+9$ as $3^{2}-x^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(3-x\right)\left(3+x\right)$$
Reorder the terms.
$$\left(-x+3\right)\left(x+3\right)$$
Rewrite the complete factored expression. Polynomial $x^{2}+9$ is not factored since it does not have any rational roots.
