1. **State the problem:** Simplify the expression $$(2p+3q)(16p^4 - 81q^4)(8p^3 - 12p^2q + 18pq^2 - 27q^3).$$ 2. **Recognize patterns:** - The term $16p^4 - 81q^4$ is a difference of squares: $$16p^4 - 81q^4 = (4p^2)^2 - (9q^2)^2 = (4p^2 - 9q^2)(4p^2 + 9q^2).$$ - The term $8p^3 - 12p^2q + 18pq^2 - 27q^3$ can be grouped and factored by grouping: $$8p^3 - 12p^2q + 18pq^2 - 27q^3 = (8p^3 - 12p^2q) + (18pq^2 - 27q^3) = 4p^2(2p - 3q) + 9q^2(2p - 3q) = (4p^2 + 9q^2)(2p - 3q).$$ 3. **Rewrite the original expression:** $$ (2p + 3q)(4p^2 - 9q^2)(4p^2 + 9q^2)(4p^2 + 9q^2)(2p - 3q).$$ 4. **Notice that $4p^2 + 9q^2$ appears twice:** $$ (2p + 3q)(4p^2 - 9q^2)(4p^2 + 9q^2)^2(2p - 3q).$$ 5. **Factor $4p^2 - 9q^2$ as difference of squares:** $$4p^2 - 9q^2 = (2p - 3q)(2p + 3q).$$ 6. **Substitute back:** $$ (2p + 3q)(2p - 3q)(2p + 3q)(4p^2 + 9q^2)^2(2p - 3q) = (2p + 3q)^2 (2p - 3q)^2 (4p^2 + 9q^2)^2.$$ 7. **Final simplified form:** $$\boxed{(2p + 3q)^2 (2p - 3q)^2 (4p^2 + 9q^2)^2}.$$