To determine which expression is equivalent to [tex]\(\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}\)[/tex], let's perform the calculations step-by-step.
### Step 1: Write down the given expression
We need to find the equivalent expression for:
[tex]\[
\frac{2a + 1}{10a - 5} \div \frac{10a}{4a^2 - 1}
\][/tex]
### Step 2: Division of fractions
Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we rewrite the expression as:
[tex]\[
\frac{2a + 1}{10a - 5} \times \frac{4a^2 - 1}{10a}
\][/tex]
### Step 3: Simplify the expressions
First, rewrite [tex]\(4a^2 - 1\)[/tex] as a difference of squares:
[tex]\[
4a^2 - 1 = (2a + 1)(2a - 1)
\][/tex]
So the expression now is:
[tex]\[
\frac{2a + 1}{10a - 5} \times \frac{(2a + 1)(2a - 1)}{10a}
\][/tex]
### Step 4: Combine the fractions
Combine the numerators and the denominators:
[tex]\[
\frac{(2a + 1) \cdot (2a + 1)(2a - 1)}{(10a - 5) \cdot 10a}
\][/tex]
### Step 5: Factor and simplify the denominator
Notice that the denominator [tex]\(10a - 5\)[/tex] can be factored out as:
[tex]\[
10a - 5 = 5(2a - 1)
\][/tex]
So our expression becomes:
[tex]\[
\frac{(2a + 1)^2 (2a - 1)}{10a \cdot 5(2a - 1)}
\][/tex]
### Step 6: Cancel common terms
The term [tex]\((2a - 1)\)[/tex] appears in both the numerator and the denominator:
[tex]\[
\frac{(2a + 1)^2 (2a - 1)}{50a (2a - 1)} = \frac{(2a + 1)^2}{50a}
\][/tex]
### Step 7: Final simplified form
The simplified expression is:
[tex]\[
\frac{(2a + 1)^2}{50a}
\][/tex]
### Step 8: Identify the matching option
Compare it to the given options:
[tex]\[
\frac{2a}{(2a-1)^2}, \quad \frac{50a}{(2a+1)^2}, \quad \frac{(2a-1)^2}{2a}, \quad \frac{(2a+1)^2}{50a}
\][/tex]
Hence, the equivalent expression is:
[tex]\[
\frac{(2a+1)^2}{50 a}
\][/tex]
So, the correct option is:
[tex]\[
\boxed{\frac{(2a + 1)^2}{50a}}
\][/tex] which corresponds to option 4.
