1. **State the problem:** Simplify or factor the quadratic expression $X^2 + 5X + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Apply to our problem:** Here, $a=1$, $b=5$, and $c=6$. We need two numbers that multiply to $1 \times 6 = 6$ and add to $5$. 4. **Find the numbers:** The numbers $2$ and $3$ satisfy this because $2 \times 3 = 6$ and $2 + 3 = 5$. 5. **Write the factored form:** Using these numbers, the factorization is $$(X + 2)(X + 3).$$ 6. **Verify by expansion:** Expanding $(X + 2)(X + 3)$ gives $$X^2 + 3X + 2X + 6 = X^2 + 5X + 6,$$ which matches the original expression. **Final answer:** $$(X + 2)(X + 3).$