Simplify (243s and 1/s) / (2/(x+1) - 3/x) = 2/3

Step 1: Combine the fractions on the left side of the equation by finding a common denominator. The common denominator for $x+1$ and $x$ is $x(x+1)$. $\frac{2}{x+1} - \frac{3}{x} = \frac{3}{2}$ $\frac{2 \cdot x}{x(x+1)} - \frac{3 \cdot (x+1)}{x(x+1)} = \frac{3}{2}$ $\frac{2x - 3(x+1)}{x(x+1)} = \frac{3}{2}$

Step 2: Distribute the $-3$ in the numerator and simplify the left side. $\frac{2x - 3x - 3}{x(x+1)} = \frac{3}{2}$ $\frac{-x - 3}{x(x+1)} = \frac{3}{2}$

Step 3: Cross-multiply to eliminate the denominators. $2(-x - 3) = 3x(x+1)$

Step 4: Expand both sides of the equation. $-2x - 6 = 3x^2 + 3x$

Step 5: Rearrange the equation into the standard quadratic form $ax^2 + bx + c = 0$. $0 = 3x^2 + 3x + 2x + 6$ $3x^2 + 5x + 6 = 0$

Step 6: Solve the quadratic equation using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a=3$, $b=5$, and $c=6$. First, calculate the discriminant $\Delta = b^2 - 4ac$. $\Delta = (5)^2 - 4(3)(6)$ $\Delta = 25 - 72$ $\Delta = -47$ Since the discriminant is negative, there are no real solutions. The solutions are complex numbers.

Step 7: Find the complex solutions. $x = \frac{-5 \pm \sqrt{-47}}{2(3)}$ $x = \frac{-5 \pm i\sqrt{47}}{6}$

The solutions are: $x_1 = \frac{-5 + i\sqrt{47}}{6}$ $x_2 = \frac{-5 - i\sqrt{47}}{6}$

The solutions are $\boxed{x = \frac{-5 \pm i\sqrt{47}}{6}}$.