Decompose $\frac{x^2 - x - 4}{x(x + 2)^2}$ into partial fractions

Question

How do I equate coefficients when decomposing the following expression into partial fractions? $$\frac{x^2 - x - 4}{x(x + 2)^2}$$

Answer 1

$$\frac{x^2-x-4}{x(x+2)^2}=\frac A{x}+\frac B{x+2}+\frac C{(x+2)^2}\implies$$

$$x^2-x-4=A(x+2)^2+Bx(x+2)+Cx$$

The above is a polynomial equality, so you can either compare coefficients of respective powers of $\,x\,$ in both sides and/or substitute some values of $\,x\,$ in both sides. For example

$$x=0\implies -4=4A\implies A=-1\\x=-2\implies 2=-2C\implies C=-1$$

$$\text{Coefficients of }\;x^2:\implies 1=A+B\implies B=2$$

and we're done...