Answer
Set $g( 4 )= \displaystyle \frac{8}{5}$
Work Step by Step
We see that $g(4) $ is not defined. We check if $\displaystyle \lim_{x\rightarrow 4}g(x)=L$ exists, and if it does, we can define the continuous extension of $g$ at $x=4$ as
$G(x)=\left\{\begin{array}{ll}
g(x) , & \text{if }x\neq 4\\
L, & \text{if }x=4
\end{array}\right.$
$L= \displaystyle \lim_{x\rightarrow 4}g(x)= \displaystyle \lim_{x\rightarrow 4}\frac{x^{2}-16}{x^{2}-3x-4}\quad $
...recognize a difference of squares, factor the trinomial
$= \displaystyle \lim_{x\rightarrow 4}\frac{(x+4)(x-4)}{(x+1)(x-4)}\quad $...cancel common term
$= \displaystyle \lim_{x\rightarrow 4}\frac{(x+4)}{(x+1)}$
... evaluate directly,
$L=\displaystyle \frac{8}{5}$
The limit exits at $x=4,$ so, if we redefine $g$,
$g(x)=\left\{\begin{array}{ll}
\dfrac{x^{2}-16}{x^{2}-3x-4} , & \text{if }x\neq 4\\
8/5, & \text{if }x=4
\end{array}\right.$
it becomes continuous.