1. **State the problem:** Factor the quadratic expression $6x^2 - 7x - 3$. 2. **Recall the factoring method:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate product and sum:** Here, $a=6$, $b=-7$, $c=-3$. So, product = $6 \times (-3) = -18$, sum = $-7$. 4. **Find two numbers:** The numbers that multiply to $-18$ and add to $-7$ are $-9$ and $2$. 5. **Rewrite the middle term:** $6x^2 - 9x + 2x - 3$. 6. **Group terms:** $(6x^2 - 9x) + (2x - 3)$. 7. **Factor each group:** $3x(2x - 3) + 1(2x - 3)$. 8. **Factor out common binomial:** $(3x + 1)(2x - 3)$. **Final answer:** $6x^2 - 7x - 3 = (3x + 1)(2x - 3)$.