Find the partial fraction expansion of5s3(s2+4)(s2+2s+2)

Step 1
The given fraction is,
${\displaystyle \frac{5s^3}{(s^2+4)(s^2+2s+2)}}$
The denominator of the fraction has two factors ${\displaystyle (s^2+4)\text{and}(s^2+2s+2)}$.
Since the quadratic equations ${\displaystyle (s^2+4)\text{and}(s^2+2s+2)}$ are not reducible, the partial fractions must have the form
${\displaystyle \frac{5s^3}{(s^2+4)(s^2+2s+2)}=\frac{As+B}{s^2+4}+\frac{Cs+D}{s^2+2s+2}}$
Step 2
Simplify further as follows.
${\displaystyle \frac{5s^3}{(s^2+4)(s^2+2s+2)}=\frac{As+B}{s^2+4}+\frac{Cs+D}{s^2+2s+2}}$
${\displaystyle 5s^3=(As+B)(s^2+2s+2)+(Cs+D)(s^2+4)}$
${\displaystyle 5s^3=s^3(B+D)+s^2(A+2B+C)+s(2A+2B+4D)+(2A+4C)}$
Equate the coefficients on both sides as follows.
${\displaystyle B+D=5}$
${\displaystyle A+2B+C=0}$
${\displaystyle 2A+2B+4D=0}$
${\displaystyle 2A+4C=0}$
Step 3
On solving the above equations, ${\displaystyle A=2,B=-8,C=3\text{and}D=4}$.
Substitute ${\displaystyle A=2,B=-8,C=3\text{and}D=4}$ in (1) as follows.
${\displaystyle \frac{5s^3}{(s^2+4)(s^2+2s+2)}=\frac{2s+(-8)}{s^2+4}+\frac{3s+4}{s^2+2s+2}}$