Rewrite $16p^{4}-81$ as $\left(4p^{2}\right)^{2}-9^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(4p^{2}-9\right)\left(4p^{2}+9\right)$$
Consider $4p^{2}-9$. Rewrite $4p^{2}-9$ as $\left(2p\right)^{2}-3^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(2p-3\right)\left(2p+3\right)$$
Rewrite the complete factored expression. Polynomial $4p^{2}+9$ is not factored since it does not have any rational roots.
