When you partial fraction something like $$\frac{x+2}{(x+3)(x+2)^2}$$ you make it $$\frac{a}{(x+3)}+\frac{b}{x+2} + \frac{c}{(x+2)^2}$$
but when you have $$\frac{10x^2+12x+20}{(x-2)(x^2+2x+4)}$$
you make it $$\frac{a}{(x-2)}+\frac{bx+c}{x^2+2x+4}$$
Why is it that you make the numerator Bx+C in the second scenario but not in the first? Even though both have quadratic denominators. Thanks
$$\frac{b}{x+2} + \frac{c}{(x+2)^2}=\frac{b(x+2)+c}{(x+2)^2}=\frac{b\,x+(2+c)}{(x+2)^2}.$$ The computation of the coefficients and the integration is easier when the fraction is written as in the left hand side.