Answer
$S$ spans $R^2$.
Work Step by Step
Assume the combination
$$a(5,0)+b(5,-4)=(0,0).$$
We have the system
\begin{align*}
5a+5b&=0\\
-4b&=0.
\end{align*}
Since the determinant of the matrix is given by
$$\left| \begin{array} {cc} 5&5\\0&-4 \end{array} \right|=-20\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0.$$
Consequently, $S$ is linearly independent and since $R^2$ has the dimension $2$ then $S$ spans $R^2$.