$y=a{{x}^{2}}+bx+c\Rightarrow {y}'=2ax+b\Rightarrow {y}''=2a$
$y+{y}'+{y}''={{x}^{2}}+x+1\Rightarrow a{{x}^{2}}+bx+c+2ax+b+2a={{x}^{2}}+x+1$
$\Rightarrow a{{x}^{2}}+\left( b+2a \right)x+\left( c+b+2a \right)={{x}^{2}}+x+1$
$\Rightarrow \left\{ \begin{matrix} a=1 \\ b+2a=1 \\ c+b+2a=1 \\ \end{matrix} \right.\Rightarrow a=1,b=-1,c=0\Rightarrow a+b+c=0$