a. \( 4 \mathrm{x}^{2}-1 \)

Let's take a fun trip through factoring! The expression \( 4\mathrm{x}^{2}-1 \) is a classic difference of squares. You can think of it as two squared numbers waiting to break free! It can be factored into \( (2\mathrm{x} - 1)(2\mathrm{x} + 1) \). So, if you ever spot something in the form of \( A^2 - B^2 \), remember that it can be factored as \( (A - B)(A + B) \). Now, this factorization has real-world applications, particularly in solving quadratic equations or simplifying expressions in algebra. For instance, by setting \( (2\mathrm{x} - 1)(2\mathrm{x} + 1) = 0 \), you can find the roots of the equation easily: \( 2\mathrm{x} = 1 \) or \( 2\mathrm{x} = -1 \), leading to the solutions \( \mathrm{x} = \frac{1}{2} \) and \( \mathrm{x} = -\frac{1}{2} \). Math in action!