1. **Problem:** Factor the quadratic expression $4k^2 + 28k + 48$. 2. **Formula and rules:** To factor a quadratic expression of the form $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Step 1: Identify coefficients:** Here, $a = 4$, $b = 28$, and $c = 48$. 4. **Step 2: Calculate $a \times c$:** $$4 \times 48 = 192$$ 5. **Step 3: Find two numbers that multiply to 192 and add to 28:** These numbers are 12 and 16 because $12 \times 16 = 192$ and $12 + 16 = 28$. 6. **Step 4: Rewrite the middle term using these numbers:** $$4k^2 + 12k + 16k + 48$$ 7. **Step 5: Group terms:** $$(4k^2 + 12k) + (16k + 48)$$ 8. **Step 6: Factor each group:** $$4k(k + 3) + 16(k + 3)$$ 9. **Step 7: Factor out the common binomial:** $$(4k + 16)(k + 3)$$ 10. **Step 8: Factor out common factor from first binomial:** $$\cancel{4}(k + 4)(k + 3)$$ 11. **Final factored form:** $$4(k + 4)(k + 3)$$ This means the quadratic $4k^2 + 28k + 48$ factors to $4(k + 4)(k + 3)$. This method is called factoring by grouping and is useful when $a \neq 1$.