Since the denominator has a linear factor $x$ and an irreducible quadratic factor $x^2 + 3$, the partial fraction decomposition will have the form:
$\frac{4x^2 - 5x + 6}{x(x^2 + 3)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 3}$
where A, B, and C are constants.
Show more…