$\frac{{\displaystyle x^4+x^3+2x^2-2x+1}}{{\displaystyle x^3+x^2}}$

$=x+\frac{{\displaystyle 2x^2-2x+1}}{{\displaystyle x^3+x^2}}$

$=x+\frac{{\displaystyle A}}{{\displaystyle x}}+\frac{{\displaystyle B}}{{\displaystyle x^2}}+\frac{{\displaystyle C}}{{\displaystyle x+1}}$ [let]

$\Rightarrow 2x^2-2x+1=Ax(x+1)+B(x+1)+C{x}^2$

By comparing the coefficients of $x^2,x$ and constant terms, we get

$A+C=2, A+B=-2$ and $B=1$

$\Rightarrow A=-3,B=1$ and $C=5$

So, $A+B+C=3$ and $P\left(x\right)=x$

So, $A+B+C=P\left(3\right)$