Answer
$$244$$
Work Step by Step
Given $$g(x, y)=x^{2}-y^{2},\ \ \ x=s^{2}+1,\ \ y=1-2s$$
Since at $s=4$, $(x,y)= (17,-7)$
$$
\frac{\partial g}{\partial \:x}=2x,\ \ \ \ \ \frac{\partial g}{\partial \:y}=-2y \\ \frac{\partial x}{\partial s}=2s,\ \ \ \ \ \ \ \ \ \ \ \frac{\partial y}{\partial s}=-2
$$
Then
\begin{align*}
\frac{\partial g }{ \partial s}&=\frac{\partial g }{ \partial x}\frac{\partial x }{ \partial s}+\frac{\partial g }{ \partial y}\frac{\partial y }{ \partial s}\\
&= 4sx+4y
\end{align*}
Hence
\begin{align*}
\frac{\partial g }{ \partial s}\bigg|_{s=4}&= 4(4)(17)+4(-7)\\
&=244
\end{align*}
