The most simple way to think about it is as follows: When you have a sum of rational functions, where denominators are polynomials, you always use the "commmon factor rule". You put in the denominator all uncommon factors and all common factors with the greatest exponent. So if you had
$\frac{1}{{x - 2}} + \frac{1}{{{{\left( {x - 2} \right)}^2}}} + \frac{1}{{x + 1}} + \frac{1}{{{{\left( {x + 1} \right)}^3}}}$
Upong merging the functions you'd get
$\frac{{A\left( x \right)}}{{{{\left( {x - 2} \right)}^2}{{\left( {x + 1} \right)}^3}}}$
Where $A$ is polynomial. But now you see you "discarded" the lower exponents. This is why you have to consider the "full" decomposition to get a satisfactory result. However, it is also useful to consider a decomposition as follows:
$\frac{{A\left( x \right)}}{{{{\left( {x - 2} \right)}^2}{{\left( {x + 1} \right)}^3}}} = \frac{{Ax + B}}{{{{\left( {x - 2} \right)}^2}}} + \frac{{C{x^2} + Dx + E}}{{{{\left( {x + 1} \right)}^3}}}$
i.e. you have to consider the polynomial of lower degree that you'd get when merging all the lower factors you have. For example in the first expression you'd get:
$\frac{1}{{x + 1}} + \frac{1}{{{{\left( {x + 1} \right)}^3}}} = \frac{{{x^2} + 2x + 2}}{{{{\left( {x + 1} \right)}^3}}}$
Hope this helped.