1. **State the problem:** Factorize the quadratic expression $x^2 + 3x + 2$. 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. Here, $a=1$, $b=3$, and $c=2$. We need two numbers that multiply to $1 \times 2 = 2$ and add to $3$. 4. The numbers are $1$ and $2$ because $1 \times 2 = 2$ and $1 + 2 = 3$. 5. Rewrite the middle term using these numbers: $$x^2 + 1x + 2x + 2$$ 6. Group terms: $$(x^2 + 1x) + (2x + 2)$$ 7. Factor each group: $$x(x + 1) + 2(x + 1)$$ 8. Factor out the common binomial: $$(x + 1)(x + 2)$$ 9. **Final answer:** The factorization of $x^2 + 3x + 2$ is $$(x + 1)(x + 2)$$.