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.

Note that $$e^x f(x) + e^x f'(x) = (e^x f(x))'$$
So if we multiply both sides of your equation by $e^x$ we have:
$$(e^x f(x))' = x^3e^x+5x^2 e^{x} + xe^x + 2e^x$$
So the solution can be found through integration:
$$e^x f(x) - e^{0}f(0) = \int_0^x (t^3 e^t + 5t^2 e^t + te^t + 2e^t) dt$$ and the right hand side can be evaluated using integration by parts. Finally you will need to multiply both sides by $e^{-x}$ to solve for $f(x)$:
$$f(x) = e^{-x} f(0) + e^{-x} \cdot \int_0^x (t^3 e^t + 5t^2 e^t + te^t + 2e^t) dt.$$
