To simplify the rational algebraic expression $\frac{2x}{x-1} + \frac{3}{x+1}$, we first need to find a common denominator and combine the terms.

1. Identify the denominators: $x-1$ and $x+1$.

2. Find the least common denominator (LCD): The LCD of $x-1$ and $x+1$ is $(x-1)(x+1)$.

3. Rewrite each fraction with the LCD: $\frac{2x}{x-1} = \frac{2x(x+1)}{(x-1)(x+1)}$ $\frac{3}{x+1} = \frac{3(x-1)}{(x+1)(x-1)}$

4. Combine the fractions: [ \frac{2x(x+1)}{(x-1)(x+1)} + \frac{3(x-1)}{(x+1)(x-1)} = \frac{2x(x+1) + 3(x-1)}{(x-1)(x+1)} \