1. **State the problem:** Factorize the quadratic equation $$6x^2 - x - 5 = 0$$. 2. **Formula and rules:** To factorize a quadratic equation of the form $$ax^2 + bx + c = 0$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate product and sum:** Here, $$a = 6$$, $$b = -1$$, and $$c = -5$$. Calculate $$a \times c = 6 \times (-5) = -30$$. We need two numbers that multiply to $$-30$$ and add to $$-1$$. 4. **Find the numbers:** The numbers are $$5$$ and $$-6$$ because $$5 \times (-6) = -30$$ and $$5 + (-6) = -1$$. 5. **Rewrite the middle term:** Rewrite $$-x$$ as $$5x - 6x$$: $$6x^2 + 5x - 6x - 5 = 0$$. 6. **Group and factor:** Group terms: $$(6x^2 + 5x) - (6x + 5) = 0$$. Factor each group: $$x(6x + 5) - 1(6x + 5) = 0$$. 7. **Factor out common binomial:** $$(x - 1)(6x + 5) = 0$$. 8. **Final factorization:** The factorized form of $$6x^2 - x - 5 = 0$$ is $$ (x - 1)(6x + 5) = 0$$. 9. **Solve for $$x$$:** Set each factor equal to zero: $$x - 1 = 0 \Rightarrow x = 1$$ $$6x + 5 = 0 \Rightarrow x = -\frac{5}{6}$$. **Answer:** $$ (x - 1)(6x + 5) = 0$$ with solutions $$x = 1$$ and $$x = -\frac{5}{6}$$.