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

Step 1 ：Given the rational equation \(\frac{3}{2 x+2}+\frac{4}{x^{2}-1}=\frac{3 x}{2(x-1)^{2}}\)

Step 2 ：To solve it, we need to find a common denominator for all the fractions. The common denominator would be the product of all the different denominators.

Step 3 ：After finding the common denominator, we multiply each term by the common denominator to get rid of the fractions.

Step 4 ：Then we simplify the equation and solve for x, which gives us the equation \(\frac{3x}{2(x-1)^{2}} = \frac{3x + 5}{2(x^{2} - 1)}\)

Step 5 ：Solving this equation gives us the solution x = -5

Step 6 ：Final Answer: The solution to the equation is \(\boxed{-5}\)