1. **State the problem:** Simplify or analyze the quadratic expression $4x^2 + 4x + 1$. 2. **Recognize the form:** This is a quadratic trinomial of the form $ax^2 + bx + c$ where $a=4$, $b=4$, and $c=1$. 3. **Check if it can be factored as a perfect square:** The formula for a perfect square trinomial is $\left(px + q\right)^2 = p^2x^2 + 2pqx + q^2$. 4. **Compare coefficients:** Here, $4x^2$ suggests $p^2 = 4$ so $p=2$. 5. **Check middle term:** $2pq = 4$ so $2 \times 2 \times q = 4$ which gives $q=1$. 6. **Check last term:** $q^2 = 1$ which matches $c=1$. 7. **Therefore, the expression factors as:** $$4x^2 + 4x + 1 = (2x + 1)^2$$ 8. **Final answer:** The quadratic expression is a perfect square and can be written as $$(2x + 1)^2$$.