\(P=\dfrac{6x^2+8x+7}{x^3-1}+\dfrac{x}{x^2+x+1}-\dfrac{6}{x-1}\)
\(=\dfrac{6x^2+8x+7}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{x}{x^2+x+1}-\dfrac{6}{x-1}\)
\(=\dfrac{6x^2+8x+7+x\left(x-1\right)-6\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{6x^2+8x+7+x^2-x-6x^2-6x-6}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}=\dfrac{1}{x-1}\)
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