1. **State the problem:** Simplify the expression $$(b - 3b^2 + 7)(2 - 3b)(5 + b)$$. 2. **Recall the distributive property:** To multiply multiple polynomials, multiply two at a time, then multiply the result by the next polynomial. 3. **Multiply the first two polynomials:** $$(b - 3b^2 + 7)(2 - 3b) = b \cdot 2 + b \cdot (-3b) - 3b^2 \cdot 2 - 3b^2 \cdot (-3b) + 7 \cdot 2 + 7 \cdot (-3b)$$ Simplify each term: $$2b - 3b^2 - 6b^2 + 9b^3 + 14 - 21b$$ Combine like terms: $$9b^3 - 9b^2 + (2b - 21b) + 14 = 9b^3 - 9b^2 - 19b + 14$$ 4. **Multiply the result by the third polynomial $(5 + b)$:** $$(9b^3 - 9b^2 - 19b + 14)(5 + b)$$ Multiply each term: $$9b^3 \cdot 5 + 9b^3 \cdot b - 9b^2 \cdot 5 - 9b^2 \cdot b - 19b \cdot 5 - 19b \cdot b + 14 \cdot 5 + 14 \cdot b$$ Simplify: $$45b^3 + 9b^4 - 45b^2 - 9b^3 - 95b - 19b^2 + 70 + 14b$$ 5. **Combine like terms:** $$9b^4 + (45b^3 - 9b^3) + (-45b^2 - 19b^2) + (-95b + 14b) + 70$$ $$= 9b^4 + 36b^3 - 64b^2 - 81b + 70$$ **Final answer:** $$9b^4 + 36b^3 - 64b^2 - 81b + 70$$