What is the product of \( (5 x+1)(5 x-1) \) ? \( \begin{array}{l}\text { A. } 25 x^{2}-1 \\ \text { B. } 10 x^{2}-1 \\ \text { C. } 25 x^{2}+10 x-1 \\ \text { D. } 10 x^{2}+10 x-1\end{array} \)

The expression \( (5x + 1)(5x - 1) \) can be simplified using the difference of squares formula, which states that \( (a + b)(a - b) = a^2 - b^2 \). Here, \( a = 5x \) and \( b = 1 \). Calculating this, we have: \[ (5x + 1)(5x - 1) = (5x)^2 - (1)^2 = 25x^2 - 1. \] Thus, the correct answer is \( \text{A. } 25 x^{2}-1 \). In real-world scenarios, this type of product can be useful when dealing with areas or determining the difference between two quantities, such as price fluctuations. For example, if you’re calculating the area of a rectangle or the effect of a price increase and decrease on profits, knowing how to expand these expressions can be incredibly beneficial! A common mistake in solving such problems is overlooking the application of the difference of squares and mistakenly distributing each term instead. Always remember to check if the expression can be simplified more elegantly first! Keep practicing, and soon you'll master these techniques!