1. **State the problem:** Simplify the expression $$(2p+3q)(16p^4 - 81q^4)(8p^3 - 12p^2q + 18pq^2 - 27q^3).$$ 2. **Identify the structure:** Notice that $16p^4 - 81q^4$ is a difference of squares because $$16p^4 = (4p^2)^2$$ and $$81q^4 = (9q^2)^2.$$ The difference of squares formula is $$a^2 - b^2 = (a - b)(a + b).$$ 3. **Factor the difference of squares:** $$16p^4 - 81q^4 = (4p^2 - 9q^2)(4p^2 + 9q^2).$$ 4. **Factor $4p^2 - 9q^2$ further:** This is also a difference of squares: $$4p^2 - 9q^2 = (2p - 3q)(2p + 3q).$$ 5. **Rewrite the original expression:** $$(2p + 3q)(2p - 3q)(2p + 3q)(4p^2 + 9q^2)(8p^3 - 12p^2q + 18pq^2 - 27q^3).$$ 6. **Combine like factors:** There are two $(2p + 3q)$ terms, so: $$(2p + 3q)^2 (2p - 3q)(4p^2 + 9q^2)(8p^3 - 12p^2q + 18pq^2 - 27q^3).$$ 7. **Factor the cubic polynomial:** The last term looks like a cubic expansion. Check if it matches the expansion of $(2p - 3q)^3$: $$(2p - 3q)^3 = 8p^3 - 36p^2q + 54pq^2 - 27q^3,$$ which is different from the given polynomial. Try factoring by grouping or recognize it as a sum/difference of cubes or other pattern. 8. **Try factoring $8p^3 - 12p^2q + 18pq^2 - 27q^3$ by grouping:** Group as $(8p^3 - 12p^2q) + (18pq^2 - 27q^3)$ $$= 4p^2(2p - 3q) + 9q^2(2p - 3q) = (4p^2 + 9q^2)(2p - 3q).$$ 9. **Substitute back:** $$(2p + 3q)^2 (2p - 3q)(4p^2 + 9q^2)(4p^2 + 9q^2)(2p - 3q) = (2p + 3q)^2 (2p - 3q)^2 (4p^2 + 9q^2)^2.$$ 10. **Final simplified form:** $$\boxed{(2p + 3q)^2 (2p - 3q)^2 (4p^2 + 9q^2)^2}.$$