1. **Problem:** How do you factor something? Here is a detailed example. 2. **Formula/rule to use:** For a quadratic like $ax^2+bx+c$, one common factoring goal is to rewrite it as $(px+q)(rx+s)$. A helpful rule is to find two numbers that multiply to $ac$ and add to $b$. 3. **Example:** Factor $x^2+7x+12$. 4. **Find two numbers:** We need two numbers that multiply to $12$ and add to $7$. The numbers are $3$ and $4$ because $3\cdot 4=12$ and $3+4=7$. 5. **Write the factors:** $$x^2+7x+12=(x+3)(x+4)$$ 6. **Check your answer:** Expand to make sure it matches. $$ (x+3)(x+4)=x^2+4x+3x+12 $$ $$ =x^2+7x+12 $$ Since it matches the original expression, the factorization is correct. 7. **Important rule:** Always look for a common factor first. If every term has something in common, factor that out before trying other methods. 8. **Final answer:** $x^2+7x+12=(x+3)(x+4)$