To simplify the given expression [tex]\(\frac{4x^2 - 8x - 5}{4x^2 - 25}\)[/tex], we need to follow these steps:
1. Factor the numerator and the denominator.
- Numerator: [tex]\(4x^2 - 8x - 5\)[/tex]
We need to find two numbers that multiply to [tex]\(4 \cdot (-5) = -20\)[/tex] and add to [tex]\(-8\)[/tex]. These numbers are [tex]\(-10\)[/tex] and [tex]\(2\)[/tex].
Rewrite the middle term ([tex]\(-8x\)[/tex]):
[tex]\[
4x^2 - 8x - 5 = 4x^2 - 10x + 2x - 5
\][/tex]
Factor by grouping:
[tex]\[
= 2x(2x - 5) + 1(2x - 5) = (2x + 1)(2x - 5)
\][/tex]
- Denominator: [tex]\(4x^2 - 25\)[/tex]
This is a difference of squares: [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex]
Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 5\)[/tex]:
[tex]\[
4x^2 - 25 = (2x)^2 - 5^2 = (2x - 5)(2x + 5)
\][/tex]
2. Simplify the fraction by canceling out common factors:
[tex]\[
\frac{4x^2 - 8x - 5}{4x^2 - 25} = \frac{(2x + 1)(2x - 5)}{(2x - 5)(2x + 5)}
\][/tex]
The factor [tex]\((2x - 5)\)[/tex] in the numerator and denominator cancel each other out:
[tex]\[
= \frac{2x + 1}{2x + 5}
\][/tex]
Thus, the expression [tex]\(\frac{4x^2 - 8x - 5}{4x^2 - 25}\)[/tex] simplifies to [tex]\(\frac{2x + 1}{2x + 5}\)[/tex].
The correct answer is:
A. [tex]\(\frac{2x + 1}{2x + 5}\)[/tex]
