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