$\begin{array}{l}
 \frac{{2{x^3} + 3x}}{{{x^4} + {x^2} + 1}} = \frac{{x + \frac{1}{2}}}{{{x^2} - x + 1}} + \frac{{x - \frac{1}{2}}}{{{x^2} + x + 1}} = \frac{1}{2}\frac{{2x + 1}}{{{x^2} - x + 1}} + \frac{1}{2}\frac{{2x - 1}}{{{x^2} + x + 1}} = \frac{1}{2}\frac{{2x - 1 + 2}}{{{x^2} - x + 1}} + \frac{1}{2}\frac{{2x + 1 - 2}}{{{x^2} + x + 1}} =  \\ 
  \\ 
  = \frac{1}{2}\frac{{2x - 1}}{{{x^2} - x + 1}} + \frac{1}{{{x^2} - x + 1}} + \frac{1}{2}\frac{{2x + 1}}{{{x^2} + x + 1}} - \frac{1}{{{x^2} + x + 1}} \\ 
 \end{array}$