1. **State the problem:** Simplify the expression $3[(3 - 6)^3 + 5 \cdot 2]$. 2. **Apply the order of operations:** First, solve inside the parentheses. 3. Calculate $3 - 6 = -3$. 4. Substitute back: $3[(-3)^3 + 5 \cdot 2]$. 5. Calculate the cube: $(-3)^3 = -27$. 6. Calculate the multiplication: $5 \cdot 2 = 10$. 7. Substitute back: $3[-27 + 10]$. 8. Simplify inside the brackets: $-27 + 10 = -17$. 9. Multiply: $3 \times -17 = -51$. **Final answer:** $-51$.