Factor Quadratic Adb23F

1. **State the problem:** We need to fully factorise the quadratic expression $$5q^2 - 17q - 12$$. 2. **Recall the factoring formula:** For a quadratic expression $$ax^2 + bx + c$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate the product and sum:** Here, $$a = 5$$, $$b = -17$$, and $$c = -12$$. Calculate $$a \times c = 5 \times (-12) = -60$$. We need two numbers that multiply to $$-60$$ and add to $$-17$$. 4. **Find the two numbers:** The pair $$-20$$ and $$3$$ works because $$-20 \times 3 = -60$$ and $$-20 + 3 = -17$$. 5. **Rewrite the middle term:** Rewrite $$-17q$$ as $$-20q + 3q$$: $$5q^2 - 20q + 3q - 12$$ 6. **Group terms:** $$(5q^2 - 20q) + (3q - 12)$$ 7. **Factor each group:** $$5q(q - 4) + 3(q - 4)$$ 8. **Factor out the common binomial:** $$(5q + 3)(q - 4)$$ **Final answer:** $$5q^2 - 17q - 12 = (5q + 3)(q - 4)$$.