Let's determine the equivalent expression for the given problem:
[tex]\[
\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}
\][/tex]
Step-by-step:
1. Understanding the problem:
We are given the expression [tex]\(\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}\)[/tex]. Recall that division by a fraction is equivalent to multiplication by its reciprocal.
2. Rewrite the division as multiplication by the reciprocal:
[tex]\[
\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1} = \frac{2a + 1}{10a - 5} \times \frac{4a^2 - 1}{10a}
\][/tex]
3. Simplify the reciprocal expression:
Notice that [tex]\(4a^2 - 1\)[/tex] is a difference of squares:
[tex]\[
4a^2 - 1 = (2a - 1)(2a + 1)
\][/tex]
Thus,
[tex]\[
\frac{2a + 1}{10a - 5} \times \frac{(2a - 1)(2a + 1)}{10a}
\][/tex]
4. Combine the fractions:
[tex]\[
\frac{(2a + 1)(2a + 1)(2a - 1)}{(10a - 5) \times 10a}
\][/tex]
5. Simplify the result:
Observe that [tex]\(10a - 5 = 5(2a - 1)\)[/tex], so the expression becomes:
[tex]\[
\frac{(2a + 1)^2 (2a - 1)}{50a (2a - 1)}
\][/tex]
The [tex]\((2a - 1)\)[/tex] term cancels out:
[tex]\[
\frac{(2a + 1)^2}{50a}
\][/tex]
Thus, the expression equivalent to [tex]\(\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}\)[/tex] is:
[tex]\[
\frac{(2a + 1)^2}{50a}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{(2a+1)^2}{50a}}
\][/tex]
