1. **State the problem:** Multiply the binomials $(-8a + 9)$ and $(3a + 12)$. 2. **Formula used:** Use the distributive property (FOIL method) for binomials: $$ (x + y)(m + n) = xm + xn + ym + yn $$ 3. **Apply the distributive property:** $$ (-8a)(3a) + (-8a)(12) + 9(3a) + 9(12) $$ 4. **Calculate each term:** $$ -8a \times 3a = -24a^2 $$ $$ -8a \times 12 = -96a $$ $$ 9 \times 3a = 27a $$ $$ 9 \times 12 = 108 $$ 5. **Combine all terms:** $$ -24a^2 - 96a + 27a + 108 $$ 6. **Simplify like terms:** $$ -24a^2 + \cancel{-96a} + 27a + 108 $$ $$ -24a^2 - \cancel{96a} + 27a + 108 $$ $$ -24a^2 - 69a + 108 $$ **Final answer:** $$ \boxed{-24a^2 - 69a + 108} $$