1. **State the problem:** Factor the quadratic expression $x^2 + 2x + 1$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, if it can be factored into $(x + m)(x + n)$, then $m + n = b$ and $mn = c$. 3. **Identify coefficients:** Here, $a=1$, $b=2$, and $c=1$. 4. **Find two numbers that add to $b=2$ and multiply to $c=1$:** These numbers are $1$ and $1$ because $1 + 1 = 2$ and $1 \times 1 = 1$. 5. **Write the factored form:** $$x^2 + 2x + 1 = (x + 1)(x + 1) = (x + 1)^2$$ 6. **Explanation:** The quadratic is a perfect square trinomial because it fits the pattern $a^2 + 2ab + b^2 = (a + b)^2$ with $a = x$ and $b = 1$. **Final answer:** $$\boxed{(x + 1)^2}$$