1. **Problem:** Factorize the expression $5x^2 - 13x - 6$ completely. 2. **Formula and rules:** To factorize a quadratic expression $ax^2 + bx + c$, find two numbers that multiply to $a \times c$ and add to $b$. 3. **Step-by-step solution:** - Given: $5x^2 - 13x - 6$ - Multiply $a$ and $c$: $5 \times (-6) = -30$ - Find two numbers that multiply to $-30$ and add to $-13$: these are $-15$ and $2$ - Rewrite the middle term using these numbers: $$5x^2 - 15x + 2x - 6$$ - Group terms: $$(5x^2 - 15x) + (2x - 6)$$ - Factor each group: $$5x(x - 3) + 2(x - 3)$$ - Factor out the common binomial: $$(x - 3)(5x + 2)$$ 4. **Final answer:** $$5x^2 - 13x - 6 = (x - 3)(5x + 2)$$ This completes the factorization.