\(\dfrac{2x+4}{x^3-1}-A=\dfrac{2}{x-1}-\dfrac{x+2}{x^2+x+1}\)
\(\Leftrightarrow\dfrac{2\left(x+2\right)}{\left(x-1\right)\left(x^2+x+1\right)}-A=\dfrac{2}{x-1}-\dfrac{x+2}{x^2+x+1}\)
\(\Leftrightarrow A=\dfrac{2\left(x+2\right)}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{2}{x-1}+\dfrac{x+2}{x^2+x+1}\)
\(\Leftrightarrow A=\dfrac{2\left(x+2\right)}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{2\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{\left(x+2\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(\Leftrightarrow A=\dfrac{2x+4-2x^2-2x-2+x^2-x+2x-2}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(\Leftrightarrow A=\dfrac{-x^2+x}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(\Leftrightarrow A=\dfrac{-x\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(\Leftrightarrow A=\dfrac{-x}{x^2+x+1}\)
