1. The problem is to solve a quadratic equation by factoring instead of using the quadratic formula. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$. 3. To solve by factoring, we look for two numbers that multiply to $ac$ and add to $b$. 4. Rewrite the middle term $bx$ using these two numbers to split it into two terms. 5. Factor by grouping: group the terms in pairs and factor out the common factor from each pair. 6. Factor out the common binomial factor. 7. Set each factor equal to zero and solve for $x$. 8. Example: Solve $x^2 + 5x + 6 = 0$ by factoring. 9. Find two numbers that multiply to $6$ and add to $5$: these are $2$ and $3$. 10. Rewrite $5x$ as $2x + 3x$: $x^2 + 2x + 3x + 6 = 0$. 11. Group terms: $(x^2 + 2x) + (3x + 6) = 0$. 12. Factor each group: $x(x + 2) + 3(x + 2) = 0$. 13. Factor out common binomial: $(x + 3)(x + 2) = 0$. 14. Set each factor to zero: $x + 3 = 0$ or $x + 2 = 0$. 15. Solve for $x$: $x = -3$ or $x = -2$. 16. Final answer: $x = -3$ or $x = -2$.