4.179. $\frac{d x}{d t}=y, \frac{d y}{d t}=-4 x-2 y$.
Solution. We form the characteristic equation

$$
\left|\begin{array}{cc}
(0-\lambda) & 1 \\
-4 & (-2-\lambda)
\end{array}\right|=2 \lambda^{2}+2 \lambda+4=0
$$

The roots of the characteristic equation are complex conjugates, the real part is negative, and the imaginary part is non-zero:

$$
\left.\lambda_{1,2}=-\frac{1}{2} \pm \frac{\sqrt{3}}{2}, \alpha=-\frac{1}{2}<0, \beta=\frac{\sqrt{3}}{2} \neq 0 \text { (case } 2 a\right) .
$$

The equilibrium point is a stable focus.