1. **State the problem:** Prove that the quadratic expression $x^2 + 5x + 6$ can be factored as $(x + 2)(x + 3)$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, factoring means finding two binomials $(x + m)(x + n)$ such that $m + n = b$ and $mn = c$. 3. **Identify coefficients:** Here, $a = 1$, $b = 5$, and $c = 6$. 4. **Find two numbers that add to $b$ and multiply to $c$:** We look for $m$ and $n$ such that $m + n = 5$ and $mn = 6$. 5. **Check possible pairs:** The pairs of factors of 6 are (1,6), (2,3), (-1,-6), (-2,-3). Among these, $2 + 3 = 5$ and $2 imes 3 = 6$. 6. **Write the factorization:** Therefore, $x^2 + 5x + 6 = (x + 2)(x + 3)$. 7. **Verify by expansion:** Expanding $(x + 2)(x + 3)$ gives $x^2 + 3x + 2x + 6 = x^2 + 5x + 6$, confirming the factorization is correct. **Final answer:** $$x^2 + 5x + 6 = (x + 2)(x + 3)$$