1. Stating the problem: Solve the quadratic equation $$5x^2 - 7x - 6 = 0$$ by factoring. 2. Multiply the coefficient of $x^2$ (which is 5) by the constant term (which is -6): $$5 \times (-6) = -30$$ 3. Find two numbers whose product is -30 and whose sum is the coefficient of $x$ (which is -7). These numbers are -10 and 3 because: $$-10 \times 3 = -30$$ $$-10 + 3 = -7$$ 4. Rewrite the middle term using these two numbers: $$5x^2 - 10x + 3x - 6 = 0$$ 5. Group the terms: $$(5x^2 - 10x) + (3x - 6) = 0$$ 6. Factor out the greatest common factor (GCF) from each group: $$5x(x - 2) + 3(x - 2) = 0$$ 7. Factor out the common binomial factor $(x - 2)$: $$(5x + 3)(x - 2) = 0$$ 8. Set each factor equal to zero and solve for $x$: $$5x + 3 = 0 \implies x = -\frac{3}{5}$$ $$x - 2 = 0 \implies x = 2$$ 9. Final answer: $$x = -\frac{3}{5} \text{ or } x = 2$$