So, first of all, we must factorize the denominator:

${x}^{3}+2{x}^{2}=(x+2)\cdot {x}^{2}$

Great. So now we write three fractions:

$\frac{A}{{x}^{2}}+\frac{B}{x}+\frac{C}{x+2}$

Eventually we conclude that

$A(x+2)+B(x+2)(x)+C({x}^{2})=5{x}^{2}+3x-2$

So now we look at what happens when x=−2:

C=12

When x=0:

A=−1And now we are missing B, but we can just pick an arbitrary number for x like... 1:

B=−1