1. **Problem:** Factor the trinomial $3x^2 - 5x - 12$ completely. 2. **Formula and Rules:** To factor a quadratic trinomial of the form $ax^2 + bx + c$ where $a \neq 1$, we use the method of factoring by grouping. This involves finding two numbers that multiply to $a \times c$ and add to $b$. 3. **Step 1: Identify coefficients:** Here, $a=3$, $b=-5$, and $c=-12$. 4. **Step 2: Calculate $a \times c$:** $$3 \times (-12) = -36$$ 5. **Step 3: Find two numbers that multiply to $-36$ and add to $-5$:** These numbers are $4$ and $-9$ because $4 \times (-9) = -36$ and $4 + (-9) = -5$. 6. **Step 4: Rewrite the middle term using these numbers:** $$3x^2 + 4x - 9x - 12$$ 7. **Step 5: Group terms:** $$(3x^2 + 4x) + (-9x - 12)$$ 8. **Step 6: Factor each group:** $$x(3x + 4) - 3(3x + 4)$$ 9. **Step 7: Factor out the common binomial:** $$(x - 3)(3x + 4)$$ 10. **Answer:** The factored form of $3x^2 - 5x - 12$ is $$\boxed{(x - 3)(3x + 4)}$$ This corresponds to option B in the SAT/ACT question. This method works by breaking down the middle term to facilitate grouping, which is a powerful technique for factoring when $a \neq 1$.