12) \( 4 n^{2}+4 n+1 \)

Did you know that the expression \( 4n^{2} + 4n + 1 \) is actually a perfect square? It can be factored elegantly as \( (2n + 1)^2 \). This means that for any integer value of \( n \), this expression will always yield a non-negative result, representing the area of a square with a side length of \( 2n + 1 \). In real-world applications, you might encounter quadratic expressions like this one in various fields such as physics or economics, where they describe trajectories, profit models, or even areas. Understanding how to manipulate and factor these expressions can be incredibly useful when solving real-life problems or optimizing certain conditions!