The problem asks to solve the quadratic equation $\frac{3}{2x+1} = \frac{x}{x-2}$.

1. Problem Description

The problem asks to solve the quadratic equation $\frac{3}{2x+1} = \frac{x}{x-2}$.

2. Solution Steps

First, we cross-multiply to eliminate the fractions:

$3(x-2) = x(2x+1)$

Expanding both sides, we get:

$3x - 6 = 2x^2 + x$

Rearrange the equation to form a standard quadratic equation $ax^2 + bx + c = 0$:

$2x^2 + x - 3x + 6 = 0$

$2x^2 - 2x + 6 = 0$

Divide the whole equation by 2:

$x^2 - x + 3 = 0$

Now we solve the quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a=1$, $b=-1$, and $c=3$.

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(3)}}{2(1)}$

$x = \frac{1 \pm \sqrt{1 - 12}}{2}$

$x = \frac{1 \pm \sqrt{-11}}{2}$

$x = \frac{1 \pm i\sqrt{11}}{2}$

3. Final Answer

$x = \frac{1 + i\sqrt{11}}{2}$ and $x = \frac{1 - i\sqrt{11}}{2}$