Answer
$ w\not\in$Span$ S$
Work Step by Step
Let $S=\{v_{1},v_{2},v_{3}\}$.
If $ w\in$Span$ S$, then there exist $a,b,c\in \mathbb{R}$
such that $w=av_{1}+bv_{2}+cv_{3}.$
$a, b,$ and $c$ are solutions (if they exist)
to a linear system with the augmented matrix
$\left[\begin{array}{lllll}
1 & 2 & 4 & | & 8\\
0 & 1 & 2 & | & 4\\
-1 & 3 & 6 & | & 7
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
\leftarrow R_{3}+R_{1}.
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & 2 & 4 & | & 8\\
0 & 1 & 2 & | & 4\\
0 & 5 & 10 & | & 15
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
\leftarrow (\frac{1}{5}R_{3}).
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & 2 & 4 & | & 8\\
0 & 1 & 2 & | & 4\\
0 & 1 & 2 & | & 3
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
\leftarrow R_{3}-R_{2}.
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & 2 & 4 & | & 8\\
0 & 1 & 2 & | & 4\\
0 & 0 & 0 & | & -1
\end{array}\right]$
(inconsistent, the last row represents the
impossible equation $0=-1)$
There are no a,b,c such that $w=av_{1}+bv_{2}+cv_{3}$,
so
$ w\not\in$Span$ S$