1. **Problem:** Factor the quadratic 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 $b$. - Rewriting the middle term using these two numbers. - Grouping terms and factoring out common factors. 3. **Step 1: Identify coefficients** - $a = 3$ - $b = -5$ - $c = -12$ 4. **Step 2: Calculate $a \times c$** $$3 \times (-12) = -36$$ 5. **Step 3: Find two numbers that multiply to $-36$ and add to $-5$** - Factors of $-36$ include $(1, -36), (-1, 36), (2, -18), (-2, 18), (3, -12), (-3, 12), (4, -9), (-4, 9), (6, -6)$ - The pair that sums to $-5$ is $4$ and $-9$ because $4 + (-9) = -5$ 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** - From $3x^2 + 4x$, factor out $x$: $x(3x + 4)$ - From $-9x - 12$, factor out $-3$: $-3(3x + 4)$ 9. **Step 7: Factor out the common binomial factor** $$(x - 3)(3x + 4)$$ 10. **Step 8: Verify by expansion** $$(x)(3x) + (x)(4) - (3)(3x) - (3)(4) = 3x^2 + 4x - 9x - 12 = 3x^2 - 5x - 12$$ **Final Answer:** The factored form of $3x^2 - 5x - 12$ is $\boxed{(x - 3)(3x + 4)}$. This corresponds to option B in the multiple-choice list.