Answer
$x^3+\dfrac{5}{3}x^2+\dfrac{5}{3}x+\dfrac{8}{3}+\dfrac{8}{3(x-1)}$
Work Step by Step
Using the long division below, then
\begin{array}{l}\require{cancel}(
x^4+\dfrac{2}{3}x^3+x
)\div(
x-1
)\\\\=
x^3+\dfrac{5}{3}x^2+\dfrac{5}{3}x+\dfrac{8}{3}+\dfrac{8}{3(x-1)}
.\end{array}