To determine which option is equivalent to [tex]\(\frac{4x^2 - 3}{2x + \sqrt{3}}\)[/tex], we need to verify each option by simplifying the expression.
First, let's rewrite the original expression for clarity:
[tex]\[
\frac{4x^2 - 3}{2x + \sqrt{3}}
\][/tex]
We need to find which of the given options equals this expression when simplified.
### Option A: [tex]\(2x - \sqrt{3}\)[/tex]
Let's check if [tex]\(2x - \sqrt{3}\)[/tex] is equivalent to [tex]\(\frac{4x^2 - 3}{2x + \sqrt{3}}\)[/tex].
1. Multiply both sides of the equation by [tex]\(2x + \sqrt{3}\)[/tex] to clear the denominator:
[tex]\[
4x^2 - 3 = (2x - \sqrt{3})(2x + \sqrt{3})
\][/tex]
2. Simplify the right-hand side:
[tex]\[
(2x - \sqrt{3})(2x + \sqrt{3}) = 2x \cdot 2x + 2x \cdot \sqrt{3} - \sqrt{3} \cdot 2x - \sqrt{3} \cdot \sqrt{3}
\][/tex]
[tex]\[
= 4x^2 + 2x\sqrt{3} - 2x\sqrt{3} - 3
\][/tex]
[tex]\[
= 4x^2 - 3
\][/tex]
The resulting expression is:
[tex]\[
4x^2 - 3
\][/tex]
This matches the left-hand side of our original equation, confirming that:
[tex]\[
2x - \sqrt{3}
\][/tex]
is indeed equivalent to [tex]\(\frac{4x^2 - 3}{2x + \sqrt{3}}\)[/tex].
### Verification of Other Options
For completeness, let's briefly check the other options:
- Option B: [tex]\(2x + \sqrt{3}\)[/tex]:
[tex]\[
\frac{4x^2 - 3}{2x + \sqrt{3}} \cdot (2x + \sqrt{3}) \neq 4x^2 - 3 \quad \text{unless simplified operations match LHS, which they don't in correct meaning.}
\][/tex]
This yields [tex]\(4x^2 + 3\)[/tex], which is incorrect.
- Option C: [tex]\(2x - 3\)[/tex]:
[tex]\[
\frac{4x^2 - 3}{2x + \sqrt{3}} \cdot (2x + \sqrt{3}) \neq 4x^2 - 3 \quad \text{unless simplified operations match LHS, which they don't in correct meaning.}
\][/tex]
- Option D: [tex]\(2x + 3\)[/tex]:
[tex]\[
\frac{4x^2 - 3}{2x + \sqrt{3}} \cdot (2x + \sqrt{3}) \neq 4x^2 - 3 \quad \text{unless simplified operations match LHS, which they don't in correct meaning.}
\][/tex]
Thus, from the consideration of the steps and verifying equivalence:
[tex]\[
\boxed{A: 2x - \sqrt{3}}
\][/tex]
is the correct answer.
