Evaluate
\frac{x^{3}-5x^{2}+9x+1}{\left(x^{2}-1\right)\left(x^{2}-x+1\right)}
Expand
\frac{x^{3}-5x^{2}+9x+1}{\left(x^{2}-1\right)\left(x^{2}-x+1\right)}
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\frac{1}{x-1}+\frac{3x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
Factor x^{2}-1.
\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{3x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
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+3x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
Since \frac{x+1}{\left(x-1\right)\left(x+1\right)} and \frac{3x+1}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.
\frac{4x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
Combine like terms in x+1+3x+1.
\frac{4x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{\left(x+1\right)\left(x^{2}-x+1\right)}
Factor 1+x^{3}.
\frac{\left(4x+2\right)\left(x^{2}-x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}-\frac{\left(3x^{2}+6x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-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^{2}-x+1\right) is \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right). Multiply \frac{4x+2}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}-x+1}{x^{2}-x+1}. Multiply \frac{3x^{2}+6x-1}{\left(x+1\right)\left(x^{2}-x+1\right)} times \frac{x-1}{x-1}.
\frac{\left(4x+2\right)\left(x^{2}-x+1\right)-\left(3x^{2}+6x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}
Since \frac{\left(4x+2\right)\left(x^{2}-x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)} and \frac{\left(3x^{2}+6x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{4x^{3}-4x^{2}+4x+2x^{2}-2x+2-3x^{3}+3x^{2}-6x^{2}+6x+x-1}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}
Do the multiplications in \left(4x+2\right)\left(x^{2}-x+1\right)-\left(3x^{2}+6x-1\right)\left(x-1\right).
\frac{x^{3}-5x^{2}+9x+1}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}
Combine like terms in 4x^{3}-4x^{2}+4x+2x^{2}-2x+2-3x^{3}+3x^{2}-6x^{2}+6x+x-1.
\frac{x^{3}-5x^{2}+9x+1}{x^{4}-x^{3}+x-1}
Expand \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right).
\frac{1}{x-1}+\frac{3x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
Factor x^{2}-1.
\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{3x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
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+3x+1}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
Since \frac{x+1}{\left(x-1\right)\left(x+1\right)} and \frac{3x+1}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.
\frac{4x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{1+x^{3}}
Combine like terms in x+1+3x+1.
\frac{4x+2}{\left(x-1\right)\left(x+1\right)}-\frac{3x^{2}+6x-1}{\left(x+1\right)\left(x^{2}-x+1\right)}
Factor 1+x^{3}.
\frac{\left(4x+2\right)\left(x^{2}-x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}-\frac{\left(3x^{2}+6x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-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^{2}-x+1\right) is \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right). Multiply \frac{4x+2}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}-x+1}{x^{2}-x+1}. Multiply \frac{3x^{2}+6x-1}{\left(x+1\right)\left(x^{2}-x+1\right)} times \frac{x-1}{x-1}.
\frac{\left(4x+2\right)\left(x^{2}-x+1\right)-\left(3x^{2}+6x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}
Since \frac{\left(4x+2\right)\left(x^{2}-x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)} and \frac{\left(3x^{2}+6x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{4x^{3}-4x^{2}+4x+2x^{2}-2x+2-3x^{3}+3x^{2}-6x^{2}+6x+x-1}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}
Do the multiplications in \left(4x+2\right)\left(x^{2}-x+1\right)-\left(3x^{2}+6x-1\right)\left(x-1\right).
\frac{x^{3}-5x^{2}+9x+1}{\left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right)}
Combine like terms in 4x^{3}-4x^{2}+4x+2x^{2}-2x+2-3x^{3}+3x^{2}-6x^{2}+6x+x-1.
\frac{x^{3}-5x^{2}+9x+1}{x^{4}-x^{3}+x-1}
Expand \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right).
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