1. **State the problem:** Factor the quadratic expression $120x^2 - 52x - 224$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate product:** $a \times c = 120 \times (-224) = -26880$. 4. **Find factors of -26880 that sum to -52:** These are 128 and -210 (since $128 + (-210) = -82$, incorrect, try again). Let's find correct pair: Factors of -26880 that add to -52 are 128 and -210 is wrong, try 128 and -210 sum is -82. Try 168 and -160: $168 + (-160) = 8$ no. Try 180 and -149: no. Try 120 and -224: sum is -104 no. Try 80 and -336: sum is -256 no. Try 96 and -280: sum is -184 no. Try 60 and -448: sum is -388 no. Try 48 and -560: sum is -512 no. Try 40 and -672: sum is -632 no. Try 32 and -840: sum is -808 no. Try 24 and -1120: sum is -1096 no. Try 16 and -1680: sum is -1664 no. Try 12 and -2240: sum is -2228 no. Try 10 and -2688: sum is -2678 no. Try 8 and -3360: sum is -3352 no. Try 6 and -4480: sum is -4474 no. Try 5 and -5376: sum is -5371 no. Try 4 and -6720: sum is -6716 no. Try 3 and -8960: sum is -8957 no. Try 2 and -13440: sum is -1338 no. Try 1 and -26880: sum is -2679 no. Since this is tedious, let's simplify the quadratic first by factoring out the GCF. 5. **Factor out GCF:** The GCF of $120, -52, -224$ is 4. $$120x^2 - 52x - 224 = 4(30x^2 - 13x - 56)$$ 6. **Now factor $30x^2 - 13x - 56$:** Calculate $a \times c = 30 \times (-56) = -1680$. 7. **Find two numbers that multiply to -1680 and add to -13:** Factors: 35 and -48 (since $35 + (-48) = -13$). 8. **Rewrite middle term:** $$30x^2 + 35x - 48x - 56$$ 9. **Group terms:** $$(30x^2 + 35x) + (-48x - 56)$$ 10. **Factor each group:** $$5x(6x + 7) - 8(6x + 7)$$ 11. **Factor out common binomial:** $$(5x - 8)(6x + 7)$$ 12. **Write final factored form:** $$4(5x - 8)(6x + 7)$$ **Answer:** The factorization of $120x^2 - 52x - 224$ is $$4(5x - 8)(6x + 7)$$.