In the beginning, you start by memorizing the rules of partial fraction decomposition, then by solving some problems, you will get the idea.
I will give you some examples
[MATH]\frac{1}{(x+1)(x-2)} = \frac{A}{x+1} + \frac{B}{x - 2}[/MATH]
[MATH]\frac{1}{(x+1)^2(x-2)} = \frac{A}{x+1} + \frac{B}{(x + 1)^2} + \frac{C}{x - 2}[/MATH]
[MATH]\frac{1}{(x+1)^3(x-2)} = \frac{A}{x+1} + \frac{B}{(x + 1)^2} + \frac{C}{(x + 1)^3} + \frac{D}{x - 2}[/MATH]
[MATH]\frac{1}{(x+1)^3(x-2)^2} = \frac{A}{x+1} + \frac{B}{(x + 1)^2} + \frac{C}{(x + 1)^3} + \frac{D}{x - 2} + \frac{E}{(x - 2)^2}[/MATH]
Did you get the idea?
Things will get interesting when we have [MATH]x^2[/MATH] inside the brackets
[MATH]\frac{1}{(x^2+1)(x-2)} = \frac{A + Bx}{x^2+1} + \frac{C}{x - 2}[/MATH]
[MATH]\frac{1}{(x^2+1)^2(x-2)} = \frac{A + Bx}{x^2+1} + \frac{C + Dx}{(x^2 + 1)^2} + \frac{E}{x - 2}[/MATH]
[MATH]\frac{1}{(x^2+1)^2(x-2)^2} = \frac{A + Bx}{x^2+1} + \frac{C + Dx}{(x^2 + 1)^2} + \frac{E}{x - 2} + \frac{F}{(x - 2)^2}[/MATH]
And so on......