ANSWER
[tex] \frac{(2a + 1)^{2}}{50a} [/tex]
EXPLANATION
The given expression is
[tex] \frac{2a + 1}{10a - 5} \div \frac{10a}{ \: 4 {a}^{2} - 1} [/tex]
Multiply by the reciprocal of the second function
[tex] \frac{2a + 1}{10a - 5} \times \frac{4 {a}^{2} - 1}{ 10a} [/tex]
Rewrite the second numerator as difference of two squares.
[tex] \frac{2a + 1}{10a - 5} \times \frac{( {2a})^{2} - {1}^{2} }{ 10a} [/tex]
Factor
[tex] \frac{2a + 1}{5(2a - 1)} \times \frac{ (2a - 1)(2a + 1) }{ 10a} [/tex]
[tex] \frac{2a + 1}{5} \times \frac{ (2a + 1) }{ 10a} [/tex]
Cancel out the common factors.
[tex] \frac{(2a + 1)(2a + 1)}{50a} [/tex]
Simplify
[tex] \frac{(2a + 1)^{2}}{50a} [/tex]
