Answer
$f(-\displaystyle \frac{2}{3})=\frac{7}{9}$
Work Step by Step
The remainder theorem:
If the polynomial f(x) is divided by $x-c$, then the remainder is $f(c).$
Divide $6x^{4}+10x^{3}+5x^{2}+x+1$
with $x+\displaystyle \frac{2}{3},\qquad (c=-\frac{2}{3})$
$\begin{array}{rrrrrrrr}
\underline{-\frac{2}{3}}| &6 & 10 & 5 & 1 &1 & & & \\
& & -4 & -4 & -\displaystyle \frac{2}{3} & -\displaystyle \frac{2}{9} & & & \displaystyle \\
& --& -- &-- &-- & -- & & & \\
&6 & 6 & 1 & \displaystyle \frac{1}{3} & \fbox{$\displaystyle \frac{7}{9}$} & & & \end{array}$
Remainder = $f(-\displaystyle \frac{2}{3})=\frac{7}{9}$
