Let's solve the expression [tex]\(\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}\)[/tex] step-by-step.
### Step 1: Understand the division
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, the expression
[tex]\[
\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}
\][/tex]
can be rewritten as
[tex]\[
\frac{2a + 1}{10a - 5} \times \frac{4a^2 - 1}{10a}.
\][/tex]
### Step 2: Substitute the reciprocal
Rewrite the expression correctly:
[tex]\[
\frac{2a + 1}{10a - 5} \times \frac{4a^2 - 1}{10a}.
\][/tex]
### Step 3: Factor expressions
Next, observe that [tex]\(4a^2 - 1\)[/tex] is a difference of squares, which can be factored as:
[tex]\[
4a^2 - 1 = (2a + 1)(2a - 1).
\][/tex]
So we can rewrite the expression as:
[tex]\[
\frac{2a + 1}{10a - 5} \times \frac{(2a + 1)(2a - 1)}{10a}.
\][/tex]
### Step 4: Combine the fractions
Multiplying the fractions, we get:
[tex]\[
\frac{(2a + 1)(2a + 1)(2a - 1)}{(10a - 5)(10a)}.
\][/tex]
### Step 5: Simplify the fraction
First, factor [tex]\(10a - 5\)[/tex] out:
[tex]\[
10a - 5 = 5(2a - 1).
\][/tex]
So our expression becomes:
[tex]\[
\frac{(2a + 1)(2a + 1)(2a - 1)}{5(2a - 1) \cdot 10a}.
\][/tex]
Now, we can cancel [tex]\(2a - 1\)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{(2a + 1)(2a + 1)}{5 \cdot 10a}.
\][/tex]
### Step 6: Simplify the remaining fraction
After cancellation, we get:
[tex]\[
\frac{(2a + 1)^2}{50a}.
\][/tex]
### Conclusion
Thus, the simplified and equivalent expression for [tex]\(\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}\)[/tex] is:
[tex]\[
\boxed{\frac{(2a + 1)^2}{50a}}
\][/tex]
So, the correct answer is:
[tex]\(\frac{(2a + 1)^2}{50a}\)[/tex].
