here is my problem:
1.for example why the form of this partial fraction should be this? $$\frac{{4{x^2}}}{{\left( {x - 1} \right){{\left( {x - 2} \right)}^2}}} = \frac{A}{{x - 1}} + \frac{B}{{x - 2}} + \frac{C}{{{{\left( {x - 2} \right)}^2}}}$$
What is wrong with $$\frac{{4{x^2}}}{{\left( {x - 1} \right){{\left( {x - 2} \right)}^2}}} = \frac{A}{{x - 1}} + \frac{B}{{{{\left( {x - 2} \right)}^2}}}$$
2.Here: $$\frac{{8{x^2} - 12}}{{x\left( {{x^2} + 2x - 6} \right)}} = \frac{A}{x} + \frac{{Bx + C}}{{{x^2} + 2x - 6}}$$
what is wrong with $$\frac{{8{x^2} - 12}}{{x\left( {{x^2} + 2x - 6} \right)}} = \frac{A}{x} + \frac{{B}}{{{x^2} + 2x - 6}}$$
I mean I dont understand why the form of Partial fraction should be like this image below and what is wrong with my examples?
As was said in comment higher just check Wikipedia. And about your example, just try to write the system for coefficients $A$ and $B$ in the case 2. You will get that $A = 8$, because the coiffecent at $x^2$ equales 8, consueqently, $-6*A = -48,$ not $-12$. It means that you can't rewrite $\frac{{8{x^2} - 12}}{{x\left( {{x^2} + 2x - 6} \right)}}$ as $ \frac{A}{x} + \frac{{B}}{{{x^2} + 2x - 6}}$