1. **State the problem:** Simplify the expression $2p(q-3)^2 + 12pq$. 2. **Recall the formula:** The square of a binomial $(a-b)^2 = a^2 - 2ab + b^2$. 3. **Expand the squared term:** $$ (q-3)^2 = q^2 - 2 \times q \times 3 + 3^2 = q^2 - 6q + 9 $$ 4. **Substitute back into the expression:** $$ 2p(q^2 - 6q + 9) + 12pq $$ 5. **Distribute $2p$ across the trinomial:** $$ 2p \times q^2 - 2p \times 6q + 2p \times 9 + 12pq = 2pq^2 - 12pq + 18p + 12pq $$ 6. **Combine like terms:** Note that $-12pq + 12pq = \cancel{-12pq} + \cancel{12pq} = 0$ 7. **Final simplified expression:** $$ 2pq^2 + 18p $$ **Answer:** $2pq^2 + 18p$