D) \( \left(2 x-\frac{1}{4}\right)^{2}= \)

To simplify \( \left(2 x-\frac{1}{4}\right)^{2} \), you can apply the formula for squaring a binomial, which states \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). Here, \( a = 2x \) and \( b = \frac{1}{4} \). So, we have: \[ (2x)^{2} - 2(2x)\left(\frac{1}{4}\right) + \left(\frac{1}{4}\right)^{2} = 4x^{2} - \frac{2}{4}(2x) + \frac{1}{16} = 4x^{2} - \frac{1}{4}x + \frac{1}{16}. \] Thus, \( \left(2 x-\frac{1}{4}\right)^{2} = 4x^{2} - \frac{1}{4}x + \frac{1}{16} \). Now, did you know that squaring binomials has roots in ancient mathematics? The Babylonians were already working with square roots over 4,000 years ago! They used geometrical methods to solve problems that closely resemble our modern algebraic techniques. Speaking of modern applications, squared expressions like this one often pop up in real-world situations like calculating areas, optimizing designs in engineering, or even in finance! Squaring a binomial can help you find the minimum space needed for a garden plot when you want to express it in quadratics.