Performing the indicated operations (Definition CVA, Definition CVSM), we obtain the vector equations
\begin{align*}
\colvector{\alpha \\ 2\alpha\\ -\alpha} + \colvector{12\\16\\8}
&=\colvector{\alpha + 12 \\ 2\alpha + 16\\ -\alpha + 8}
=\colvector{-1\\0\\4}
\end{align*}
Thus, if a solution $\alpha$ exists, by Definition CVE then $\alpha$ must satisfy the three equations:
\begin{align*}
\alpha + 12 &= -1\\
2\alpha + 16 &= 0\\
-\alpha + 8 &= 4
\end{align*}
which leads to $\alpha = -13$, $\alpha = -8$ and $\alpha = 4$.
Since $\alpha$ cannot simultaneously have three different values, there is no solution to the original vector equation.