Answer
$S$ spans $R^3$.
Work Step by Step
Assume the combination
$$a(4,7,3)+b(-1,2,6)+c(2,-3,5)=(0,0,0).$$
We have the system
\begin{align*}
4a-b+2c&=0\\
7a+2b-3c&=0\\
3a+6b+5c&=0
\end{align*}
Since the determinant of the matrix is given by
$$\left| \begin{array} {cc} 4&-1&2\\7&-2&-3\\3&6&5 \end{array} \right|=228\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0, \quad c=0.$$
Consequently, $S$ is linearly independent and since $R^3$ has the dimension $3$ then $S$ spans $R^3$.
