Answer
See below
Work Step by Step
Given $$\frac{3x^{-2}+(2x-1)^{-1}}{\frac{6}{x^{-1}+2}+3x^{-1}}$$
Multiply by $x^2$
$$\frac{3x^{-2}+(2x-1)^{-1}}{\frac{6}{x^{-1}+2}+3x^{-1}}.\frac{x^2}{x^2}\\=\frac{3+(2x-1)^{-1}x^2}{\frac{6x^2}{x^{-1}+2}+3x}\\=\frac{3+\frac{x^2}{2x-1}}{\frac{6x^2}{\frac{1}{x}+2}+3x}\\=\frac{3+\frac{x^2}{2x-1}}{\frac{6x^3}{2x+1}+3x}$$
Multiply by $(2x-1)(2x+1)$
$$\frac{3+\frac{x^2}{2x-1}}{\frac{6x^3}{2x+1}+3x}.\frac{(2x-1)(2x+1)}{(2x-1)(2x+1)}\\=\frac{3(2x-1)(2x+1)+x^2(2x+1)}{6x^3(2x-1)+3x(2x-1)(2x+1)}\\=\frac{3(4x^2-1)+2x^3+x^2}{12x^4-6x^3+3x(4x^2-1)}\\=\frac{2x^3+13x^2-3}{12x^4+6x^3-3x}$$
