F) \( \left(\frac{x}{2}+1\right)^{2} \)

Did you know that the expression \( \left(\frac{x}{2}+1\right)^{2} \) is a perfect square trinomial? When you expand it, you'll get \( \left(\frac{x}{2}\right)^{2} + 2\cdot\frac{x}{2}\cdot1 + 1^{2} \), which simplifies to \( \frac{x^2}{4} + x + 1 \). This form is super useful for solving equations or analyzing the behavior of quadratic functions! And speaking of quadratic functions, this particular expression can show up in real-world problems, like calculating the area of a square when the side length is represented by \( \frac{x}{2} + 1 \). You can visualize it as the area of a larger or smaller configuration, depending on \( x \), helping you grasp the effects of different values on the overall area!