Answer
Remainder: $0$
$x+4$ is a factor of $f(x).$
Work Step by Step
The Remainder Theorem says that the remainder when a function $f(x)$ is divided by $(x-r)$ is $f(r)$.
Hence, the remainder when $f(x)$ si divided by $x+4$ is:
\begin{align*}
f(-4)&=(-4)^6-16\cdot(-4)^4+(-4)^2-16\\
&=4096-4096+16-16\\
&=0
\end{align*}
The factor theorem says that if $f(c)=0$, then $(x-c)$ is a factor of $f(x)$ and vice versa.
With $f(-4)=0$, then $x+4$ is a factor of $f(x)$ by the factor theorem.