Mixed Exercise between polynomials and derivatives

Exercise:

$if\,$  $f\left( x \right)+f'\left( x \right)={{x}^{3}}+{{x}^{2}}+x+1$

Determine the expression for $f\left( x \right)$

Solution:  Let $f\left( x \right)\in \mathbb{R}\left[ x \right]$ then $f\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d$ where $a,b,c,d\in \mathbb{R}$

So $f'\left( x \right)=3a{{x}^{2}}+2bx+c$ Now adding $f'\left( x \right)\,\,\And \,\,f\left( x \right)$ to get :

$f\left( x \right)+f'\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d+3a{{x}^{2}}+2bx+c={{x}^{3}}+{{x}^{2}}+x+1$

$\Rightarrow f\left( x \right)+f'\left( x \right)=a{{x}^{3}}+{{x}^{2}}\left( b+3a \right)+x\left( c+2b \right)+\left( d+c \right)={{x}^{3}}+{{x}^{2}}+x+1$

So $a=1\,\,,\,\,\,b+3a=1\,\,,\,\,c+2b=1\,\,,\,\,d+c=1$ $\Rightarrow a=1\,\,\,,\,\,b=-2\,\,\,,\,\,c=5\,\,\,,\,\,d=-4$

Thus $f\left( x \right)={{x}^{3}}-2{{x}^{2}}+5x-4$