(x+22)/((x+4)(x-2))=A/(x+4)+B/(x-2) where A and B have to be found.
Write the RHS using the LCD (x+4)(x-2):
(x+22)/((x+4)(x-2))=(A(x-2)+B(x+4))/((x+4)(x-2)).
Since the denominators on each side of equals are the same we can simply equate the numerators:
x+22=A(x-2)+B(x+4),
x+22=Ax-2A+Bx+4B.
There is only one x on the LHS and on the RHS we have Ax+Bx. Therefore A+B=1.
We have the constant 22 on the LHS and on the RHS we have -2A+4B=22.
We have a pair of simultaneous equations which needs to be solved.
Multiply the first equation by 2: 2A+2B=2 and add to the second equation to eliminate A:
6B=24, so B=4. Since A+B=1, A+4=1, A=-3.
Now we can substitute these values using the first line of this answer:
(x+22)/((x+4)(x-2))=A/(x+4)+B/(x-2)=-3/(x+4)+4/(x-2).
The RHS is the solution in partial fractions.