1. The problem is to factor a quadratic expression using the product-sum method. 2. The product-sum method involves finding two numbers that multiply to the product of the quadratic coefficient and the constant term, and add to the middle coefficient. 3. For a quadratic $ax^2 + bx + c$, find two numbers $m$ and $n$ such that $m \times n = a \times c$ and $m + n = b$. 4. Rewrite the middle term $bx$ as $mx + nx$. 5. Factor by grouping the four terms into two binomials. 6. Example: Factor $x^2 + 5x + 6$. 7. Here, $a=1$, $b=5$, $c=6$. 8. Find $m$ and $n$ such that $m \times n = 1 \times 6 = 6$ and $m + n = 5$. 9. The numbers are 2 and 3. 10. Rewrite $x^2 + 5x + 6$ as $x^2 + 2x + 3x + 6$. 11. Group terms: $(x^2 + 2x) + (3x + 6)$. 12. Factor each group: $x(x + 2) + 3(x + 2)$. 13. Factor out the common binomial: $(x + 2)(x + 3)$. 14. Final answer: $x^2 + 5x + 6 = (x + 2)(x + 3)$.