$\displaystyle \frac{2x-3}{x(x^2+1)} \Rightarrow 2x-3 = \frac{A}{x} + \frac{Bx+C}{x^2+1} $ The question is, how does the numerator on $\displaystyle x^2 + 1 $ equal $\displaystyle Bx + C $ ?
If the denominator of the rational expression has an irreducible quadratic factor, then you have to account for the possible "size" of the numerator. If the denominator contains a degree-two factor, then the numerator might not be just a constant; it might be of degree one. So you would deal with a quadratic factor in the denominator by including a linear expression in the numerator.
quoted from this site ...
Partial-Fraction Decomposition: Repeated and Irreducible Factors
You need to combine the two fractions such that there will be no $\displaystyle x^2$ term in the resulting numerator
$\displaystyle \frac{A}{x}+\frac{Bx+C}{x^2+1}=\frac{x^2+1}{x^2+1} \ \frac{A}{x}+\frac{x}{x}\ \frac{Bx+C}{x^2+1}$
$\displaystyle =\frac{Ax^2+A+Bx^2+Cx}{x\left(x^2+1\right)}$
This allows the $\displaystyle x^2$ terms to cancel as $\displaystyle B=-A,$
Also $\displaystyle C=2,\ A=-3$