We work from the definition of the pre-image, Definition PI. Setting
\begin{equation*}
\lt{S}{\colvector{a\\b\\c}}=\colvector{-2\\5\\3}
\end{equation*}
we arrive at a system of three equations in three variables, with an augmented matrix that we row-reduce in a search for solutions,
\begin{equation*}
\begin{bmatrix}
1 & -2 & -1 & -2 \\
3 & -1 & 2 & 5 \\
1 & 1 & 2 & 3
\end{bmatrix}
\rref
\begin{bmatrix}
\leading{1} & 0 & 1 & 0 \\
0 & \leading{1} & 1 & 0 \\
0 & 0 & 0 & \leading{1}
\end{bmatrix}
\end{equation*}
With a leading 1 in the last column, this system is inconsistent (Theorem RCLS), and there are no values of $a$, $b$ and $c$ that will create an element of the pre-image. So the preimage is the empty set.
We work from the definition of the pre-image, Definition PI. Setting
\begin{equation*}
\lt{S}{\colvector{a\\b\\c}}=\colvector{-5\\5\\7}
\end{equation*}
we arrive at a system of three equations in three variables, with an augmented matrix that we row-reduce in a search for solutions,
\begin{equation*}
\begin{bmatrix}
1 & -2 & -1 & -5 \\
3 & -1 & 2 & 5 \\
1 & 1 & 2 & 7
\end{bmatrix}
\rref
\begin{bmatrix}
\leading{1} & 0 & 1 & 3 \\
0 & \leading{1} & 1 & 4 \\
0 & 0 & 0 & 0
\end{bmatrix}
\end{equation*}
The solution set to this system, which is also the desired pre-image, can be expressed using the vector form of the solutions (Theorem VFSLS)
\begin{equation*}
\preimage{S}{\colvector{-5\\5\\7}}
=\setparts{\colvector{3\\4\\0}+c\colvector{-1\\-1\\1}}{c\in\complex{}}
=\colvector{3\\4\\0}+\spn{\set{\colvector{-1\\-1\\1}}}
\end{equation*}
Does the final expression for this set remind you of Theorem KPI?