1. **State the problem:** We are given the demand function $p(q) = 140 - 3q$ and the total cost function $C(q) = 12q^2 + 12q + 500$. We need to find the profit function $P(q)$. 2. **Recall the profit function formula:** Profit is revenue minus cost. $$P(q) = R(q) - C(q)$$ where revenue $R(q) = p(q) \times q$. 3. **Calculate the revenue function:** $$R(q) = p(q) \times q = (140 - 3q)q = 140q - 3q^2$$ 4. **Write the profit function:** $$P(q) = R(q) - C(q) = (140q - 3q^2) - (12q^2 + 12q + 500)$$ 5. **Simplify the profit function:** $$P(q) = 140q - 3q^2 - 12q^2 - 12q - 500$$ $$P(q) = 140q - 15q^2 - 12q - 500$$ $$P(q) = (140q - 12q) - 15q^2 - 500$$ $$P(q) = 128q - 15q^2 - 500$$ **Final answer:** $$P(q) = -15q^2 + 128q - 500$$