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\frac{3\left(x^{2}+x+1\right)}{\left(x-1\right)\left(x^{2}+x+1\right)}-\frac{3\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 x-1 and x^{2}+x+1 is \left(x-1\right)\left(x^{2}+x+1\right). Multiply \frac{3}{x-1} times \frac{x^{2}+x+1}{x^{2}+x+1}. Multiply \frac{3}{x^{2}+x+1} times \frac{x-1}{x-1}.
\frac{3\left(x^{2}+x+1\right)-3\left(x-1\right)}{\left(x-1\right)\left(x^{2}+x+1\right)}
Since \frac{3\left(x^{2}+x+1\right)}{\left(x-1\right)\left(x^{2}+x+1\right)} and \frac{3\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{3x^{2}+3x+3-3x+3}{\left(x-1\right)\left(x^{2}+x+1\right)}
Do the multiplications in 3\left(x^{2}+x+1\right)-3\left(x-1\right).
\frac{3x^{2}+6}{\left(x-1\right)\left(x^{2}+x+1\right)}
Combine like terms in 3x^{2}+3x+3-3x+3.
\frac{3x^{2}+6}{x^{3}-1}
Expand \left(x-1\right)\left(x^{2}+x+1\right).