1. **State the problem:** Factorize the quadratic expression $6x^2 - 17x - 3$. 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, $a = 6$, $b = -17$, and $c = -3$. So, $a \times c = 6 \times (-3) = -18$. 4. **Find two numbers that multiply to $-18$ and add to $-17$:** These numbers are $-18$ and $1$ because $-18 \times 1 = -18$ and $-18 + 1 = -17$. 5. **Rewrite the middle term using these numbers:** $$6x^2 - 18x + 1x - 3$$ 6. **Group terms:** $$(6x^2 - 18x) + (1x - 3)$$ 7. **Factor each group:** $$6x(x - 3) + 1(x - 3)$$ 8. **Factor out the common binomial:** $$(6x + 1)(x - 3)$$ **Final answer:** The factorization of $6x^2 - 17x - 3$ is $$\boxed{(6x + 1)(x - 3)}$$.