#### Answer

$f(-\displaystyle \frac{2}{3})=\frac{7}{9}$

#### Work Step by Step

The last step in synthetic division tells us how to interpret the last row:
7. Use the numbers in the last row to write the quotient, plus the remainder above the divisor.
The degree of the first term of the quotient is one less than the degree of the first term of the dividend.
The final value in this row is the remainder.
The Remainder Theorem states:
If the polynomial $f(x)$ is divided by $x-c$, then the remainder is $f(c)$.
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Our plan: perform synthetic division (dividing with $x-c)$
and the last number in the last row is $f(c)$.
Arrange the powers,
$(6x^{4}+10x^{3}+5x^{2}+x+1)\displaystyle \div(x+\frac{2}{3})$
$\begin{array}{lllllll}
\underline{-\frac{2}{3}}| & 6 & 10 & 5 & 1 & 1 & \\
& & -4 & -4 & -\frac{2}{3} & -\frac{2}{9} & \\
& -- & -- & -- & -- & -- & \\
& 6 & 6 & 1 & \frac{1}{3} & |\overline{ \frac{7}{9} } &
\end{array}$
The remainder, $f(-\displaystyle \frac{2}{3})=\frac{7}{9}$