1. **State the problem:** Solve the quadratic expression $x^2 + 3x + 2$ by factoring. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. 3. **Apply to the problem:** Here, $a=1$, $b=3$, and $c=2$. We need two numbers that multiply to $1 \times 2 = 2$ and add to $3$. 4. **Find the numbers:** The numbers are $1$ and $2$ because $1 \times 2 = 2$ and $1 + 2 = 3$. 5. **Rewrite the middle term:** $$x^2 + 1x + 2x + 2$$ 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 factored form of $x^2 + 3x + 2$ is $$(x + 1)(x + 2)$$.