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\frac{1}{2\left(x-1\right)}-\frac{1}{4\left(x+1\right)}+\frac{1}{2x+2}
Factor 2x-2. Factor 4x+4.
\frac{2\left(x+1\right)}{4\left(x-1\right)\left(x+1\right)}-\frac{x-1}{4\left(x-1\right)\left(x+1\right)}+\frac{1}{2x+2}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-1\right) and 4\left(x+1\right) is 4\left(x-1\right)\left(x+1\right). Multiply \frac{1}{2\left(x-1\right)} times \frac{2\left(x+1\right)}{2\left(x+1\right)}. Multiply \frac{1}{4\left(x+1\right)} times \frac{x-1}{x-1}.
\frac{2\left(x+1\right)-\left(x-1\right)}{4\left(x-1\right)\left(x+1\right)}+\frac{1}{2x+2}
Since \frac{2\left(x+1\right)}{4\left(x-1\right)\left(x+1\right)} and \frac{x-1}{4\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2x+2-x+1}{4\left(x-1\right)\left(x+1\right)}+\frac{1}{2x+2}
Do the multiplications in 2\left(x+1\right)-\left(x-1\right).
\frac{x+3}{4\left(x-1\right)\left(x+1\right)}+\frac{1}{2x+2}
Combine like terms in 2x+2-x+1.
\frac{x+3}{4\left(x-1\right)\left(x+1\right)}+\frac{1}{2\left(x+1\right)}
Factor 2x+2.
\frac{x+3}{4\left(x-1\right)\left(x+1\right)}+\frac{2\left(x-1\right)}{4\left(x-1\right)\left(x+1\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4\left(x-1\right)\left(x+1\right) and 2\left(x+1\right) is 4\left(x-1\right)\left(x+1\right). Multiply \frac{1}{2\left(x+1\right)} times \frac{2\left(x-1\right)}{2\left(x-1\right)}.
\frac{x+3+2\left(x-1\right)}{4\left(x-1\right)\left(x+1\right)}
Since \frac{x+3}{4\left(x-1\right)\left(x+1\right)} and \frac{2\left(x-1\right)}{4\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.
\frac{x+3+2x-2}{4\left(x-1\right)\left(x+1\right)}
Do the multiplications in x+3+2\left(x-1\right).
\frac{3x+1}{4\left(x-1\right)\left(x+1\right)}
Combine like terms in x+3+2x-2.
\frac{3x+1}{4x^{2}-4}
Expand 4\left(x-1\right)\left(x+1\right).