1. **State the problem:** Simplify the expression $3x - 4x^2 - 9x + 12$. 2. **Group like terms:** Combine the terms with $x^2$, $x$, and constants separately. 3. **Rewrite the expression:** $$3x - 4x^2 - 9x + 12 = -4x^2 + (3x - 9x) + 12$$ 4. **Simplify the $x$ terms:** $$3x - 9x = -6x$$ 5. **Final simplified expression:** $$-4x^2 - 6x + 12$$ 6. **Factor if possible:** Factor out the greatest common factor (GCF) from all terms. 7. **Find GCF:** The GCF of $-4x^2$, $-6x$, and $12$ is $-2$. 8. **Factor out $-2$:** $$-4x^2 - 6x + 12 = -2(2x^2 + 3x - 6)$$ 9. **Check if quadratic inside can be factored:** Try to factor $2x^2 + 3x - 6$. 10. **Use the AC method:** Multiply $a$ and $c$: $2 \times (-6) = -12$. Find two numbers that multiply to $-12$ and add to $3$: $6$ and $-2$. 11. **Rewrite middle term:** $$2x^2 + 6x - 2x - 6$$ 12. **Group terms:** $$(2x^2 + 6x) + (-2x - 6)$$ 13. **Factor each group:** $$2x(x + 3) - 2(x + 3)$$ 14. **Factor out common binomial:** $$(2x - 2)(x + 3)$$ 15. **Factor out 2 from $(2x - 2)$:** $$2(x - 1)(x + 3)$$ 16. **Include the $-2$ factored out earlier:** $$-2 \times 2 (x - 1)(x + 3) = -4 (x - 1)(x + 3)$$ **Final answer:** $$-4 (x - 1)(x + 3)$$