For $\vect{b}$ to be an element of $W=\spn{S}$ there must be a linear combination of the vectors in $S$ that equals $\vect{b}$ (Definition SSCV). The existence of such scalars is equivalent to the linear system $\linearsystem{A}{\vect{b}}$ being consistent, where $A$ is the matrix whose columns are the vectors from $S$ (Theorem SLSLC).
\begin{align*}
\begin{bmatrix}
1 & 1 & 0 & 1 & 2\\
2 & 0 & 1 & 1 & 1\\
0 & 3 & 0 & 2 & 2\\
2 & 1 & 2 & 0 & 0
\end{bmatrix}
&\rref
\begin{bmatrix}
\leading{1} & 0 & 0 & 0 & 3/2\\
0 & \leading{1} & 0 & 0 & 1\\
0 & 0 & \leading{1} & 0 & -3/2 \\
0 & 0 & 0& \leading{1} & -1/2
\end{bmatrix}
\end{align*}
So by Theorem RCLS the system is consistent, which indicates that $\vect{b}$ is in the subspace $W$.