Find $f(x)$ from the differential equation $f(x)+{f}'(x) = x^{3} +5x^{2} +x +2$ find $f(x)$ from
$f(x)+{f}'(x) = x^{3} +5x^{2} +x +2$.
I tried to impose $f(x)$ and $f'(x)$ but i can not solve it.

If f(x) is a polynomial,
Let $f(x)=ax^3+bx^2+cx+d$
Then $f'(x)=3ax^2+2bx+c$
Therefore $f(x)+f'(x)=ax^3+(3a+b)x^2+(2b+c)x+(c+d)$
$a=1, b=2, c=-3,d=5$
EDIT
If $f(x)$ is partially polynomial.
Let $f(x)=g(x)+h(x)$, where $g(x)$ is polynomial and $h(x)$ is non-polynomial,
$f'(x)=g'(x)+h'(x)$
when $f(x)+f'(x)=[g(x)+g'(x)]+[h'(x)+h(x)]=G(x)+H(x)$
where $G(x)=x^3+5x^2+x+2$
And for any  ${x:h(x)+h'(x)=0}$, the non-polynomial part is cancelled.
e.g. $h(x)=e^{-x}, h'(x)=-e^{-x}$
$f(x)=x^3+5x^2+x+2+h(x)$
