1. **State the problem:** Solve the equation $$15 \cdot 8(3b + 1) = 4(7b + 3) - 9$$ for $b$. 2. **Rewrite the equation:** The problem is to find $b$ such that $$15 \times 8(3b + 1) = 4(7b + 3) - 9$$. 3. **Simplify both sides:** $$15 \times 8(3b + 1) = 120(3b + 1) = 360b + 120$$ $$4(7b + 3) - 9 = 28b + 12 - 9 = 28b + 3$$ 4. **Set the equation:** $$360b + 120 = 28b + 3$$ 5. **Bring all terms to one side:** $$360b - 28b = 3 - 120$$ $$332b = -117$$ 6. **Solve for $b$:** $$b = \frac{-117}{332}$$ 7. **Check if $b=1$ satisfies the equation:** Substitute $b=1$: Left side: $$15 \times 8(3(1) + 1) = 15 \times 8(4) = 15 \times 32 = 480$$ Right side: $$4(7(1) + 3) - 9 = 4(10) - 9 = 40 - 9 = 31$$ Since $480 \neq 31$, $b=1$ is not a solution. **Final answer:** $$b = \frac{-117}{332}$$