1. **Problem:** Factor the expression $$12y^2 + 3y - 9$$. 2. **Step 1: Identify common factors.** The terms are $$12y^2$$, $$3y$$, and $$-9$$. The greatest common factor (GCF) is 3. 3. **Step 2: Factor out the GCF.** $$12y^2 + 3y - 9 = 3(4y^2 + y - 3)$$ 4. **Step 3: Factor the quadratic inside the parentheses.** We want two numbers that multiply to $$4 \times (-3) = -12$$ and add to $$1$$ (the coefficient of $$y$$). These numbers are 4 and -3. 5. **Step 4: Rewrite the middle term using these numbers.** $$3(4y^2 + 4y - 3y - 3)$$ 6. **Step 5: Group terms and factor each group.** $$3[(4y^2 + 4y) - (3y + 3)]$$ $$= 3[4y(y + 1) - 3(y + 1)]$$ 7. **Step 6: Factor out the common binomial factor.** $$3(y + 1)(4y - 3)$$ **Final answer:** $$3(y + 1)(4y - 3)$$