To find the quotient of the rational expressions [tex]\(\frac{2x}{4x+3} \div \frac{x-1}{2x}\)[/tex], we need to follow the steps of division for fractions, which involves multiplying by the reciprocal of the divisor. Let's break it down step-by-step:
1. Identify the given expressions:
[tex]\[\frac{2x}{4x+3}\][/tex]
[tex]\(\div\)[/tex]
[tex]\[\frac{x-1}{2x}\][/tex]
2. Rewrite the division as multiplication by the reciprocal of the divisor:
[tex]\[\frac{2x}{4x+3} \times \frac{2x}{x-1}\][/tex]
3. Multiply the numerators together and the denominators together:
The numerators are:
[tex]\[2x \times 2x = 4x^2\][/tex]
The denominators are:
[tex]\[(4x + 3) \times (x - 1) = 4x(x - 1) + 3(x - 1)\][/tex]
4. Distribute in the denominator:
[tex]\[4x(x - 1) = 4x^2 - 4x\][/tex]
[tex]\[3(x - 1) = 3x - 3\][/tex]
5. Combine the distributed parts of the denominator:
[tex]\[ (4x^2 - 4x) + (3x - 3) = 4x^2 - 4x + 3x - 3 = 4x^2 - x - 3 \][/tex]
6. Write the complete expression after multiplication:
[tex]\[\frac{4x^2}{4x^2 - x - 3}\][/tex]
Thus, the quotient of the rational expressions [tex]\(\frac{2x}{4x+3} \div \frac{x-1}{2x}\)[/tex] is
[tex]\[\boxed{\frac{4x^2}{4x^2 - x - 3}}\][/tex]
This matches option B given in the problem.
