1. **State the problem:** Factor the trinomial $$4x^2 - 13x + 10$$ into the form $$(4x - a)(x - b)$$ where $a$ and $b$ are numbers to be found. 2. **Recall the factoring method:** For a trinomial $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate the product and sum:** Here, $a=4$, $b=-13$, and $c=10$. So, multiply $4 \times 10 = 40$. 4. **Find two numbers that multiply to 40 and add to -13:** These numbers are -8 and -5 because $-8 \times -5 = 40$ and $-8 + (-5) = -13$. 5. **Rewrite the middle term using these numbers:** $$4x^2 - 8x - 5x + 10$$ 6. **Group terms and factor each group:** $$ (4x^2 - 8x) + (-5x + 10) $$ $$ 4x(x - 2) - 5(x - 2) $$ 7. **Factor out the common binomial:** $$ (4x - 5)(x - 2) $$ 8. **Check by expanding:** $$ (4x - 5)(x - 2) = 4x^2 - 8x - 5x + 10 = 4x^2 - 13x + 10 $$ **Final answer:** $$(4x - 5)(x - 2)$$