Define the following

$\frac{x+2 x^{2}+3 x^{4}-x^{5}}{2 x}$

$=\frac{x}{2 x}+\frac{2 x^{2}}{2 x}+\frac{3 x^{4}}{2 x}-\frac{x^{5}}{2 x}$

$=\frac{1}{2}+x+\frac{3}{2} x^{3}-\frac{1}{2} x^{4}$