\(\frac{x+2}{x-1}\left(\frac{x^3}{2x+2}+1\right)-\frac{8x+7}{2x^2-2}Đkxđ:x\ne\pm1\)
\(=\frac{x+2}{x-1}.\frac{x^3+2x+2}{2\left(x+1\right)}-\frac{8x+7}{2\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x^4+2x^3+2x^2-2x-3}{2\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x^4-2x^2+2x^3-2x+3x^2-3}{2\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x^2\left(x^2-1\right)+2x\left(x^2-1\right)+3\left(x^2-1\right)}{2\left(x-1\right)\left(x+1\right)}\)
\(=\frac{\left(x+1\right)\left(x-1\right)\left(x^2+2x+3\right)}{2\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x^2+2x+3}{2}\)
Ta có: \(x^2+2x+3=x^2+2x+1+2\)
\(=\left(x+1\right)^2+2\ge2\forall x\)
Vậy giá trị của biểu thức luôn dương \(\forall x\ne\pm1\)
