Which of the following is the quotient of the rational expressions shown below? (x+2)/x-1 / 3/2x A. (3x+6)/2x^2-2x B. (2x^2+4x)/3x-3 C. 3/x-1 D. 4x/-3

To solve the problem of finding the quotient of the given rational expressions, we need to divide the first rational expression by the second. Let's go through the steps:

1. Break Down the Problem

   • We have two rational expressions: $\frac{x+2}{x-1}$ and $\frac{3}{2x}$.
   • We need to find the quotient of these expressions, which means dividing the first expression by the second.

2. Relevant Concepts

   • Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, $\frac{\frac{x+2}{x-1}}{\frac{3}{2x}}$ is equivalent to $\frac{x+2}{x-1} \times \frac{2x}{3}$.

3. Analysis and Detail

   • Multiply the numerators: $(x+2) \times 2x = 2x(x+2) = 2x^2 + 4x$.
   • Multiply the denominators: $(x-1) \times 3 = 3(x-1) = 3x - 3$.

4. Verify and Summarize

   • The resulting expression is $\frac{2x^2 + 4x}{3x - 3}$.
   • This matches option B.

Final Answer

The quotient of the rational expressions is $\frac{2x^2 + 4x}{3x - 3}$, which corresponds to option B.