Answer
$f(-\frac{2}{3})=\frac{7}{9}$
Work Step by Step
Step 1. Based on the Remainder Theorem, the value of $f(c)$ can be obtained as the remainder of $f(x)$ divided by $x-c$.
Step 2. The coefficients of the dividend $f(x)$ can be identified in order as $\{6,10,5,1,1\}$ and the divisor is $x+\frac{2}{3}$; use synthetic division as shown in the figure to get the quotient and the remainder.
Step 3. We can identify the result as $f(x)=(6x^3+6x^2+x+\frac{1}{3})(x+\frac{2}{3})+\frac{7}{9}$; thus we have $f(-\frac{2}{3})=\frac{7}{9}$.
