1. **State the problem:** We need to factor the quadratic expression $10x^2 + 13x - 3$. 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, $a = 10$, $b = 13$, and $c = -3$. Calculate $a \times c = 10 \times (-3) = -30$. We need two numbers that multiply to $-30$ and add to $13$. 4. **Find the two numbers:** The numbers are $15$ and $-2$ because $15 \times (-2) = -30$ and $15 + (-2) = 13$. 5. **Rewrite the middle term:** Rewrite $13x$ as $15x - 2x$: $$10x^2 + 15x - 2x - 3$$ 6. **Group terms and factor each group:** Group as $(10x^2 + 15x) + (-2x - 3)$. Factor each group: $$5x(2x + 3) - 1(2x + 3)$$ 7. **Factor out the common binomial:** $$ (5x - 1)(2x + 3) $$ **Final answer:** $$10x^2 + 13x - 3 = (5x - 1)(2x + 3)$$