1. **Problem statement:** We want to factor and solve a simple quadratic function, for example, $x^2 + 5x + 6 = 0$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c = 0$, find two numbers that multiply to $ac$ and add to $b$. 3. **Step-by-step factoring:** For $x^2 + 5x + 6$, $a=1$, $b=5$, $c=6$. 4. Find two numbers that multiply to $1 \times 6 = 6$ and add to $5$. These are $2$ and $3$. 5. Rewrite the middle term: $x^2 + 2x + 3x + 6 = 0$. 6. Group terms: $(x^2 + 2x) + (3x + 6) = 0$. 7. Factor each group: $x(x + 2) + 3(x + 2) = 0$. 8. Factor out common binomial: $(x + 3)(x + 2) = 0$. 9. **Solve:** Set each factor equal to zero: $$x + 3 = 0 \Rightarrow x = -3$$ $$x + 2 = 0 \Rightarrow x = -2$$ 10. **Answer:** The solutions are $x = -3$ and $x = -2$.