ta có: \(\frac{5-x}{x^2+2}+\left\lbrack\frac{5}{x-3}-\left(\frac{2}{4-x}+\frac{x-3}{x^2+2}\right)\right\rbrack-\left(\frac{2}{x-4}+\frac{x-3}{x+1}\right)\)
\(=\frac{5-x}{x^2+2}+\frac{5}{x-3}-\frac{2}{4-x}-\frac{x-3}{x^2+2}-\frac{2}{x-4}-\frac{x-3}{x-1}\)
\(=\frac{5-x}{x^2+2}-\frac{x-3}{x^2+2}+\frac{5}{x-3}-\frac{x-3}{x-1}+\frac{2}{x-4}-\frac{2}{x-4}\)
\(=\frac{5-x-x+3}{x^2+2}+\frac{5}{x-3}-\frac{x-3}{x-1}=\frac{-2x+8}{x^2+2}+\frac{5\left(x-1\right)-\left(x-3\right)^2}{\left(x-3\right)\left(x-1\right)}\)
\(=\frac{-2x+8}{x^2+2}+\frac{5x-5-x^2+6x-9}{\left(x-3\right)\left(x-1\right)}=\frac{-2x+8}{x^2+2}+\frac{-x^2+11x-14}{\left(x-3\right)\left(x-1\right)}\)
\(=\frac{\left(-2x+8\right)\left(x^2-4x+3\right)+\left(-x^2+11x-14\right)\left(x^2+2\right)}{\left(x^2+2\right)\left(x-3\right)\left(x-1\right)}\)
\(=\frac{-2x^3+8x^2-6x+8x^2-32x+24-x^4-2x^2+11x^3+22x-14x^2-28}{\left(x^2+2\right)\left(x-1\right)\left(x-3\right)}\)
\(=\frac{-x^4+9x^3-16x-4}{\left(x^2+2\right)\left(x-1\right)\left(x-3\right)}\)
