Complex conjugate roots

Characteristic equation.
$$r^2 + 4r + 5 = 0.$$
The discriminant is $\Delta = 16 - 20 = -4 = (2i)^2$, so the roots are
$$r_{1,2} = \frac{-4 \pm 2i}{2} = -2 \pm i.$$

General solution.
$$y(x) = e^{-2x}(A\cos x + B\sin x), \qquad A,B \in \mathbb{R}.$$

Remark. This is the oscillatory (pseudo-periodic) case.