Question

how do you i do the partial fraction for 1/(s^2)(s^2 +2s +3)

Answer 1

What you typed is really just

(s^2 +2s +3)/s^2

which is

= 1 + 2/s + 3/s^2

I don't know if this is what you needed.

Answer 2

If, however, you meant

1/((s^2)(s^2 +2s +3)) then that is
A/s + B/s^2 + (Cs+D)/(s^2+2s+3)
= A(s*(s^2+2s+3)) + B(s^2+2s+3) + (Cs+D)s^2
= As^3+2As^2+3As + Bs^2+2Bs+3B + Cs^3+Ds^2
= (A+C)s^3 + (2A+B+D)s^2 + (3A+2B)s + 3B

all over s^2(s^2+2s+3)

So, to make the two sides equal, we need
A+C = 0
2A+B+D = 0
3A+2B = 0
3B = 1

Solve that and you get
B = 1/3
A = -2/9
C = 2/9
D = 1/9

So, the partial fraction decomposition is

-2/9s + 1/3s^2 + (2s-1)/9(s^2+2s+3)

Answer 3

oops the first one should be s^2 +1

Answer 4

so, fix it and follow the steps.

If you enter your formula at wolframalpha.com, it will show the decomposition, so you can check your work. Watch to be sure you include necessary parentheses. wolframalpha will show how it interprets your input.