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\frac{1}{x-1}+\frac{2x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
Factor x^{2}-1.
\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{2x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right). Multiply \frac{1}{x-1} times \frac{x+1}{x+1}.
\frac{x+1+2x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
Since \frac{x+1}{\left(x-1\right)\left(x+1\right)} and \frac{2x+1}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.
\frac{3x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
Combine like terms in x+1+2x+1.
\frac{3x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{\left(x-1\right)\left(-x-1\right)}
Factor 1-x^{2}.
\frac{3x+2}{\left(x-1\right)\left(x+1\right)}-\frac{-\left(3x^{2}+5x+1\right)}{\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 \left(x-1\right)\left(x+1\right) and \left(x-1\right)\left(-x-1\right) is \left(x-1\right)\left(x+1\right). Multiply \frac{3x^{2}+5x+1}{\left(x-1\right)\left(-x-1\right)} times \frac{-1}{-1}.
\frac{3x+2-\left(-\left(3x^{2}+5x+1\right)\right)}{\left(x-1\right)\left(x+1\right)}
Since \frac{3x+2}{\left(x-1\right)\left(x+1\right)} and \frac{-\left(3x^{2}+5x+1\right)}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{3x+2+3x^{2}+5x+1}{\left(x-1\right)\left(x+1\right)}
Do the multiplications in 3x+2-\left(-\left(3x^{2}+5x+1\right)\right).
\frac{8x+3+3x^{2}}{\left(x-1\right)\left(x+1\right)}
Combine like terms in 3x+2+3x^{2}+5x+1.
\frac{8x+3+3x^{2}}{x^{2}-1}
Expand \left(x-1\right)\left(x+1\right).
\frac{1}{x-1}+\frac{2x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
Factor x^{2}-1.
\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{2x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right). Multiply \frac{1}{x-1} times \frac{x+1}{x+1}.
\frac{x+1+2x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
Since \frac{x+1}{\left(x-1\right)\left(x+1\right)} and \frac{2x+1}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.
\frac{3x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{1-x^{2}}
Combine like terms in x+1+2x+1.
\frac{3x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+5x+1}{\left(x-1\right)\left(-x-1\right)}
Factor 1-x^{2}.
\frac{3x+2}{\left(x-1\right)\left(x+1\right)}-\frac{-\left(3x^{2}+5x+1\right)}{\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 \left(x-1\right)\left(x+1\right) and \left(x-1\right)\left(-x-1\right) is \left(x-1\right)\left(x+1\right). Multiply \frac{3x^{2}+5x+1}{\left(x-1\right)\left(-x-1\right)} times \frac{-1}{-1}.
\frac{3x+2-\left(-\left(3x^{2}+5x+1\right)\right)}{\left(x-1\right)\left(x+1\right)}
Since \frac{3x+2}{\left(x-1\right)\left(x+1\right)} and \frac{-\left(3x^{2}+5x+1\right)}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{3x+2+3x^{2}+5x+1}{\left(x-1\right)\left(x+1\right)}
Do the multiplications in 3x+2-\left(-\left(3x^{2}+5x+1\right)\right).
\frac{8x+3+3x^{2}}{\left(x-1\right)\left(x+1\right)}
Combine like terms in 3x+2+3x^{2}+5x+1.
\frac{8x+3+3x^{2}}{x^{2}-1}
Expand \left(x-1\right)\left(x+1\right).