I am trying to find the partial fraction decomposition of $\dfrac{4s^2 - 5s + 2}{s^2(s^2 +9)}$ into something of the form $A\dfrac{1}{s} + B\dfrac{1}{s^2} + C\dfrac{1}{s^2+9} + D\dfrac{s}{s^2 + 9}$.
I am unable to factor the numerator into linear terms and how to factor the denominator into linear terms in a useful way is unclear to me since if I just break $s^2$ into $s \cdot s$ then it doesn't seem very helpful. Thank you in advance.
$$\dfrac{4s^2 - 5s + 2}{s^2(s^2 +9)}=\frac{As+B}{s^2}+\frac{Cs+D}{s^2+9}$$ Multiply both sides by $s^2(s^2-9):$ $${4s^2 - 5s + 2}=({As+B})({s^2}+9)+{(Cs+D)}s^2$$ Combine terms: $$4s^2-5s+2=(A+C)s^3+(B+D)s^2+(9A)s+9B$$ Solve for coefficients: $$9B=2\implies B=\frac 29$$ $$\left(\frac 29+D\right)=4\implies D=\frac{34}9$$ $$9A=-5\implies A=-\frac{5}9$$ $$-\frac 59+C=0\implies C=\frac 59$$