1. **State the problem:** Factor the quadratic expression $2m^2 - 3m - 2$ completely. 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, $a=2$, $b=-3$, and $c=-2$. So, $a \times c = 2 \times (-2) = -4$. We need two numbers that multiply to $-4$ and add to $-3$. 4. **Find the pair:** The numbers are $-4$ and $1$ because $-4 \times 1 = -4$ and $-4 + 1 = -3$. 5. **Rewrite the middle term:** $$2m^2 - 4m + 1m - 2$$ 6. **Group terms:** $$(2m^2 - 4m) + (1m - 2)$$ 7. **Factor each group:** $$2m(m - 2) + 1(m - 2)$$ 8. **Factor out the common binomial:** $$(2m + 1)(m - 2)$$ **Final answer:** $$\boxed{(2m + 1)(m - 2)}$$