Partial fractions help for laplace transforms

I need some help please

A(s^2+9)+B(s^2+1)=s^2+10
so when i set s=3i i get -8B= S^2+10 so do you just match coefficients, then i would get B=-5/4
This is wrong though, the book says the answers is B =-1/8 and A=9/8, am i going about this completely wrong
or does anybody have any resources for advanced partial fractions.
Thanks in advance

If you set s = 3i, then you don't get −8B = s² + 10
What you do get is −8B = −9 + 10 = 1 ------> B = −1/8

And when you set s = i, then you get
8A = −1 + 10 = 9 -----> A = 9/8

Note that there are usually 2 different ways to find partial fractions:
Matching coefficients (by expanding left side) OR by substitution.

You were attempting substitution.
DO NOT try matching coefficients if you are using this approach.

I have no clue as to what you are doing setting s = 3i ... and I suspect you don't either.

The equations for coefficient matching are:

A+B = 1
9A+B = 10 and the solution is: A= 9/8 and B= -1/8

distrubute the A and B then set the s^2 terms equal and the constants equal

As^2+9A+Bs^2+B=s^2+10

A+B=1
9A+B=10

L2-L1>L2
8A=9
A=9/8 now put this into either and you get B=-1/8

Just match coefficients of like terms to get a 2x2 system in A and B

A + B = 1
9A + B = 10
-- - - - - - - - -
8A = 9
A = 9/8
B = -1/8