1. **State the problem:** Simplify the expression $$74 \cdot (3x+2)(3x-9) + (5x^2 - 7x^3)$$. 2. **Recall the distributive property and FOIL method:** To multiply two binomials, use FOIL (First, Outer, Inner, Last). 3. **Multiply the binomials:** $$(3x+2)(3x-9) = 3x \cdot 3x + 3x \cdot (-9) + 2 \cdot 3x + 2 \cdot (-9)$$ $$= 9x^2 - 27x + 6x - 18$$ $$= 9x^2 - 21x - 18$$ 4. **Multiply the result by 74:** $$74 \cdot (9x^2 - 21x - 18) = 74 \cdot 9x^2 - 74 \cdot 21x - 74 \cdot 18$$ $$= 666x^2 - 1554x - 1332$$ 5. **Add the remaining polynomial:** $$666x^2 - 1554x - 1332 + 5x^2 - 7x^3$$ 6. **Combine like terms:** $$-7x^3 + (666x^2 + 5x^2) - 1554x - 1332$$ $$= -7x^3 + 671x^2 - 1554x - 1332$$ **Final answer:** $$\boxed{-7x^3 + 671x^2 - 1554x - 1332}$$