1. **State the problem:** Divide the expression $16p^4 - 81q^4$ by $2p + 3q$. 2. **Formula and rules:** We will use polynomial division and the difference of squares formula. Recall that $a^2 - b^2 = (a - b)(a + b)$. 3. **Rewrite the numerator:** Notice that $16p^4 = (4p^2)^2$ and $81q^4 = (9q^2)^2$, so $$16p^4 - 81q^4 = (4p^2)^2 - (9q^2)^2$$ 4. **Apply difference of squares:** $$= (4p^2 - 9q^2)(4p^2 + 9q^2)$$ 5. **Factor further:** The term $4p^2 - 9q^2$ is also a difference of squares: $$4p^2 - 9q^2 = (2p)^2 - (3q)^2 = (2p - 3q)(2p + 3q)$$ 6. **Rewrite the numerator fully factored:** $$16p^4 - 81q^4 = (2p - 3q)(2p + 3q)(4p^2 + 9q^2)$$ 7. **Divide by $2p + 3q$:** $$\frac{16p^4 - 81q^4}{2p + 3q} = \frac{(2p - 3q)(2p + 3q)(4p^2 + 9q^2)}{2p + 3q}$$ 8. **Cancel common factor:** $$= (2p - 3q)(4p^2 + 9q^2)$$ **Final answer:** $$\boxed{(2p - 3q)(4p^2 + 9q^2)}$$