To factor the expression [tex]\(4x^2 - \frac{1}{4}\)[/tex], let's follow these steps:
1. Identify the form of the given expression:
[tex]\[
4x^2 - \frac{1}{4}
\][/tex]
2. Notice that this expression is a difference of squares:
[tex]\[
4x^2 - \frac{1}{4} = (2x)^2 - \left(\frac{1}{2}\right)^2
\][/tex]
3. Recall the factoring formula for the difference of squares, which is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
4. Applying the difference of squares formula to our expression:
[tex]\[
a = 2x \quad \text{and} \quad b = \frac{1}{2}
\][/tex]
Thus,
[tex]\[
4x^2 - \frac{1}{4} = (2x - \frac{1}{2})(2x + \frac{1}{2})
\][/tex]
5. Verify if our factored expression matches any of the given choices. Let’s compare:
- (A) [tex]\(\left(2x + \frac{1}{2}\right)\left(2x - \frac{1}{2}\right)\)[/tex]
- (B) [tex]\(\left(x + \frac{1}{2}\right)\left(2x - \frac{1}{2}\right)\)[/tex]
- (C) [tex]\(\left(x + \frac{1}{2}\right)\left(x - \frac{1}{2}\right)\)[/tex]
- (D) [tex]\(\left(2x + \frac{1}{2}\right)\left(x - \frac{1}{2}\right)\)[/tex]
The factored form [tex]\((2x - \frac{1}{2})(2x + \frac{1}{2})\)[/tex] matches choice (A).
Therefore, the correct answer is:
[tex]\[
\boxed{\text{A}}
\][/tex]
