1. **State the problem:** Factor the polynomial $$-12a^4 + 9a^3 - 3a^2$$ completely, including factoring out a negative number or a GCF with a negative coefficient. 2. **Identify the greatest common factor (GCF):** Look at the coefficients and the variable powers. - Coefficients: -12, 9, -3. The GCF of 12, 9, and 3 is 3. - Variables: $$a^4, a^3, a^2$$. The smallest power is $$a^2$$. 3. **Include the negative sign in the GCF:** Since the first term is negative, factor out $$-3a^2$$. 4. **Write the factored form:** $$-12a^4 + 9a^3 - 3a^2 = -3a^2(\cancel{\frac{12}{-3}}4a^{4-2} - \cancel{\frac{9}{-3}}(-3)a^{3-2} + \cancel{\frac{3}{-3}}(-1)a^{2-2})$$ 5. **Simplify inside the parentheses:** $$-3a^2(-4a^2 + 3a - 1)$$ 6. **Final answer:** $$\boxed{-3a^2(-4a^2 + 3a - 1)}$$ This is the completely factored form with a negative GCF factored out.