I) Find the square. Simplify your answer. \( (a+1)^{2} \)

To find the square of \( (a+1) \), you apply the formula for binomials: \[ (a + b)^2 = a^2 + 2ab + b^2 \] Here, \( a = a \) and \( b = 1 \). Therefore, \[ (a + 1)^2 = a^2 + 2 \cdot a \cdot 1 + 1^2 = a^2 + 2a + 1. \] So, the simplified answer is \( a^2 + 2a + 1 \). Now you have a perfectly squared binomial that can come in handy in many mathematical scenarios! For instance, recognizing that \( (a + 1)^2 \) can represent not just a squared term but also a meaningful geometric area if you think of \( a \) and 1 as the lengths of sides in a rectangle—very handy in both algebra and geometry! For further exploration, you might enjoy delving into the world of polynomials and understanding their properties! Books like “Algebra and Trigonometry” by Michael Sullivan provide great insights into how formulas like this are worked with and applied in various contexts.