1. **State the problem:** Factor completely the quadratic expression $5x^2 - 6x - 8$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate the product and sum:** Here, $a=5$, $b=-6$, $c=-8$. Calculate $a \times c = 5 \times (-8) = -40$. We need two numbers that multiply to $-40$ and add to $-6$. 4. **Find the pair:** The numbers are $4$ and $-10$ because $4 \times (-10) = -40$ and $4 + (-10) = -6$. 5. **Rewrite the middle term:** $$5x^2 - 6x - 8 = 5x^2 + 4x - 10x - 8$$ 6. **Group terms:** $$(5x^2 + 4x) + (-10x - 8)$$ 7. **Factor each group:** $$x(5x + 4) - 2(5x + 4)$$ 8. **Factor out the common binomial:** $$(5x + 4)(x - 2)$$ **Final answer:** $$5x^2 - 6x - 8 = (5x + 4)(x - 2)$$