The values of k for which the points $A(1,0,3),B(-1,3,4),C(1,2,1)\text{ and }D(k,2,5)$ are coplanar

Let $\mathbf{a}=(1,0,3),\mathbf{b}=(-1,3,4),\mathbf{c}=(1,2,1)$ and

$\mathbf{d}=(k,2,5).\text{ Since }\mathbf{A},\mathbf{B},\mathbf{C}\text{ and }\mathbf{D}$ are coplanar, we have

$\begin{array}{l}\left[\mathbf{dbc}\right]+\left[\mathbf{dca}\right]+\left[\mathbf{dab}\right]=\left[\begin{array}{lll}\mathbf{a}& \mathbf{b}& \mathbf{c}\end{array}\right]\\ \Rightarrow \left|\begin{array}{ccc}k& 2& 5\\ -1& 3& 4\\ 1& 2& 1\end{array}\right|+\left|\begin{array}{ccc}k& 2& 5\\ 1& 2& 1\\ 1& 0& 3\end{array}\right|+\left|\begin{array}{ccl}k& 2& 5\\ 1& 0& 3\\ -1& 3& 4\end{array}\right|=\left|\begin{array}{ccc}1& 0& 3\\ -1& 3& 4\\ 1& 2& 1\end{array}\right|\\ \Rightarrow -5k-15+6k-14-9k+1=-20\\ \Rightarrow -8k-28=-20\Rightarrow k=-1\end{array}$