Which expression is a difference of squares with a factor of $5x - 8$?
To determine which expression is a difference of squares and has a factor of \(5x - 8\), we first recall the identity for the difference of squares: \[A^2 - B^2 = (A - B)(A + B).\] We are looking for an expression of the form \(A^2 - B^2\) where one of the factors is \(5x - 8\). This means that \(A = 5x\) and \(B = 8\), so our expression must take the form \((5x)^2 - 8^2 = 25x^2 - 64.\)
Now we evaluate each of the given options:
By comparing our derived target expression \(25x^2 - 64\) with the provided choices, we confirm that the correct expression is indeed \(25x^2 - 64\). Therefore, the difference of squares that includes \(5x - 8\) as a factor is \(\boxed{25x^2 - 64}\).
In architecture or construction, you might need to determine the area of a specific section of a floor plan. If you have a square room of side length \(5x\) and you remove a square cutout of side length \(8\) from the corner, the remaining area can be expressed as \((5x)^2 - 8^2\). Understanding how to factor this as \((5x - 8)(5x + 8)\) helps in simplifying design calculations and material estimations, making it easier to plan for flooring or tile layouts that need to fit around such features.