To determine the quotient of the rational expressions [tex]\(\frac{4x}{2x-1} \div \frac{3x+2}{x+5}\)[/tex], we need to follow these steps:
1. Rewrite the division as multiplication: The division of fractions can be rewritten as the multiplication of the first fraction by the reciprocal of the second fraction.
[tex]\[
\frac{4x}{2x-1} \div \frac{3x+2}{x+5} = \frac{4x}{2x-1} \times \frac{x+5}{3x+2}
\][/tex]
2. Multiply the numerators together:
[tex]\[
\text{Numerator: } 4x \times (x + 5) = 4x(x + 5) = 4x^2 + 20x
\][/tex]
3. Multiply the denominators together:
[tex]\[
\text{Denominator: } (2x-1) \times (3x+2)
\][/tex]
Now we need to expand the product of the denominators:
[tex]\[
(2x-1)(3x+2) = 2x \cdot 3x + 2x \cdot 2 - 1 \cdot 3x - 1 \cdot 2
\][/tex]
[tex]\[
(2x-1)(3x+2) = 6x^2 + 4x - 3x - 2
\][/tex]
[tex]\[
(2x-1)(3x+2) = 6x^2 + x - 2
\][/tex]
4. Combine the results to form the quotient:
[tex]\[
\frac{4x^2 + 20x}{6x^2 + x - 2}
\][/tex]
So, the quotient of [tex]\(\frac{4x}{2x-1} \div \frac{3x+2}{x+5}\)[/tex] simplifies to:
[tex]\[
\frac{4x^2 + 20x}{6x^2 + x - 2}
\][/tex]
Looking at the answer choices provided, the correct one is:
[tex]\[
\boxed{\frac{4x^2 + 20x}{6x^2 + x - 2}}
\][/tex]
