Answer
$S$ spans $R^2$.
Work Step by Step
Assume the combination of two vectors only of $S$
$$a(-1,4)+b(1,1)=(0,0).$$
We have the system
\begin{align*}
-a+b&=0\\
4a+b&=0.
\end{align*}
Since the determinant of the matrix is given by
$$\left| \begin{array} {cc} -1&1\\4&1\end{array} \right|=-5\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0.$$
Consequently, the vectors $(-1,4),(1,1)$ are linearly independent and since $R^2$ has the dimension $2$ then they span $R^2$ and hence $S$ spans $R^2$.