1. **Problem Statement:** Factor the trinomial 4x^2 + 16x + 12. 2. **Formula and Rules:** To factor a trinomial of the form $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. 3. **Step 1:** Calculate $ac = 4 \times 12 = 48$. 4. **Step 2:** Find factor pairs of 48 that add to 16: 4 and 12. 5. **Step 3:** Rewrite the middle term: $4x^2 + 4x + 12x + 12$. 6. **Step 4:** Factor by grouping: $$4x^2 + 4x + 12x + 12 = (4x^2 + 4x) + (12x + 12)$$ $$= 4x(x + 1) + 12(x + 1)$$ 7. **Step 5:** Factor out the common binomial: $$= (4x + 12)(x + 1)$$ 8. **Step 6:** Factor out common factor 4 from first binomial: $$= \cancel{(4)}(x + 3)(x + 1)$$ 9. **Final answer:** $$4(x + 3)(x + 1)$$ --- 1. **Problem Statement:** Factor the trinomial 2x^2 - 16x + 30. 2. **Calculate:** $ac = 2 \times 30 = 60$. 3. **Find factors of 60 that add to -16:** -10 and -6. 4. **Rewrite:** $2x^2 - 10x - 6x + 30$. 5. **Group:** $(2x^2 - 10x) + (-6x + 30) = 2x(x - 5) - 6(x - 5)$. 6. **Factor:** $(2x - 6)(x - 5)$. 7. **Factor out 2:** $\cancel{(2)}(x - 3)(x - 5)$. 8. **Final answer:** $$2(x - 3)(x - 5)$$ --- 1. **Problem Statement:** Factor the trinomial 3x^2 + 12x - 63. 2. **Calculate:** $ac = 3 \times (-63) = -189$. 3. **Find factors of -189 that add to 12:** 21 and -9. 4. **Rewrite:** $3x^2 + 21x - 9x - 63$. 5. **Group:** $3x(x + 7) - 9(x + 7)$. 6. **Factor:** $(3x - 9)(x + 7)$. 7. **Factor out 3:** $\cancel{(3)}(x - 3)(x + 7)$. 8. **Final answer:** $$3(x - 3)(x + 7)$$ --- 1. **Problem Statement:** Factor the trinomial 6x^2 + 12x - 48. 2. **Calculate:** $ac = 6 \times (-48) = -288$. 3. **Find factors of -288 that add to 12:** 24 and -12. 4. **Rewrite:** $6x^2 + 24x - 12x - 48$. 5. **Group:** $6x(x + 4) - 12(x + 4)$. 6. **Factor:** $(6x - 12)(x + 4)$. 7. **Factor out 6:** $\cancel{(6)}(x - 2)(x + 4)$. 8. **Final answer:** $$6(x - 2)(x + 4)$$