The trick is:

If you have a factor of in the denominator you have a term in your decomposition.

If you have a factor of in the denominator you have a term in your decomposition.

etc.

If you have factor of in your denominator then you have terms in your decomposition.

If you have factor of in your denominator then you have terms in your decomposition.

etc.

As an example, let's do the first one:

We have linear factors of x - 2 and x - 3 in the denominator, so we use

How do we find A and B? My favorite method (there are many) is to add the fractions on the LHS:

and match the coefficients of powers of x on both sides of the equation.

So we know that

and

I get that A = -2 and B = 3, thus

It might take longer than some of the other tricks, but I find it keeps me more organized with what I am doing.

Try the other 2 in this set. I got:

and

-Dan