first write the equation in matrix form:
\begin{equation*}\textbf{x}'=\begin{pmatrix}\hphantom{-}-1 & -4\\\hphantom{-}1 &-1\end{pmatrix}\textbf{x}\ . \end{equation*}
find eigenvalues:
\begin{equation*} r^2 - trace(A) + (ad - bc)= r^2+ 2r + 5 = 0\implies r_1= -1 + 2i, r_2=-1 -2i\end{equation*}
then, find eigenvectors, which are conjugated
\begin{equation*} \begin{pmatrix} -1 - r & \hphantom{-}-4\\ \hphantom{-}1 &-1 -r\end{pmatrix}\begin{pmatrix}\mathbf{\xi}_1\\\mathbf{\xi}_2\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix} \end{equation*}
find the two conjugate eigenvectors
\begin{equation*}\mathbf{\xi}^1 =\begin{pmatrix}2\\-i\end{pmatrix}
\mathbf{\xi}^2 =\begin{pmatrix}2\\i\end{pmatrix}\end{equation*}
therefore
\begin{equation*}\mathbf{x}(t)= C_1e^{-t}\begin{pmatrix}2\cos(2t)\\\sin(t) \end{pmatrix}+ C_2e^{-t}\begin{pmatrix}-2\sin(2t)\\\cos(t) \end{pmatrix} \end{equation*}
the attachment is the phase portrait generated by PPLANE
(spiral point, stable)
