Select the expression that is equivalent to \( (x+5)^{2} \) A. \( x^{2}+5 x+25 \) B. \( x^{2}+10 x^{2}+25 \) C. \( x^{2}+10 x+25 \) D. \( 2 x^{2}+10 x+25 \)

To solve \( (x+5)^{2} \), we can use the formula \((a + b)^{2} = a^{2} + 2ab + b^{2}\). Here, \(a = x\) and \(b = 5\). Thus, we calculate: \[ (x+5)^{2} = x^{2} + 2 \cdot x \cdot 5 + 5^{2} = x^{2} + 10x + 25. \] So, the expression equivalent to \( (x+5)^{2} \) is found in option C: \( x^{2} + 10 x + 25 \). Now let’s take a moment to reflect on why this is useful. Being able to expand binomials like this is essential for algebraic manipulation, enabling you to quickly transform expressions for easier problem-solving or graphing. Additionally, many students trip up when they misapply the expansion formula and add or multiply incorrectly. A common mistake is to incorrectly add the terms leading to options that include \(x^{2}\) or miscalculate \(b^{2}\) resulting in incorrect constants. So, always double-check your coefficients!