Answer
$12$.
Work Step by Step
By the Remainder Theorem, if $f(x)$ is divided by $x-c$, then the remainder is $f(c)$.
So if $f(x)$ is divided by $x+2$, then the remainder is $f(-2)$, the value of $c$ is in this case $-2$.
Use synthetic division
$\begin{matrix}
-2) &1&-2&-11&5&34 \\
& &-2&8&6 &-22\\
& --&--&--& --&--\\
& 1&-4&-3&11&12
\end{matrix}$
The remainder $12$, is the value of $f(-2)$.
Check
Replace $x$ with $-2$ in the given function.
$f(-2)=(-2)^4-2(-2)^3-11(-2)^2+5(-2)+34$
Simplify.
$f(-2)=16+16-44-10+34$
$f(-2)=66-54$
$f(-2)=12\checkmark$.