Answer
$\frac{x^2-8x-1}{2(x+1)^2(x-1)}$.
Work Step by Step
Factor each term of the given expression.
$\frac{x-1}{x^2+2x+1}-\frac{3}{2x-2}+\frac{x}{x^2-1}=\frac{x-1}{(x+1)(x+1)}-\frac{3}{2(x-1)}+\frac{x}{(x+1)(x-1)}$
The LCD is $=2(x+1)(x+1)(x-1)$.
Multiply each numerator and denominator by the extra factor required to form the LCD.
$=\frac{x-1}{(x+1)(x+1)}\times \frac{2(x-1)}{2(x-1)}-\frac{3}{2(x-1)}\times \frac{(x+1)(x+1)}{(x+1)(x+1)}+\frac{x}{(x+1)(x-1)}\times \frac{2(x+1)}{2(x+1)}$
Simplify.
$=\frac{2(x-1)(x-1)}{2(x+1)(x+1)(x-1)}-\frac{3(x+1)(x+1)}{2(x+1)(x+1)(x-1)}+\frac{2x(x+1)}{2(x+1)(x+1)(x-1)}$
Add all the numerators.
$=\frac{2(x-1)(x-1)-3(x+1)(x+1)+2x(x+1)}{2(x+1)(x+1)(x-1)}$
Use distributive property in the numerator.
$=\frac{2(x^2-2x+1)-3(x^2+2x+1)+2x^2+2x}{2(x+1)(x+1)(x-1)}$
Clear the parentheses.
$=\frac{2x^2-4x+2-3x^2-6x-3+2x^2+2x}{2(x+1)(x+1)(x-1)}$
Simplify.
$=\frac{x^2-8x-1}{2(x+1)(x+1)(x-1)}$.
or we can write.
$=\frac{x^2-8x-1}{2(x+1)^2(x-1)}$.
