4. If the complex numbers $z_{1}$ and $z_{2}$ satisfy the equations $\left|z_{1}+2 i\right|=2$ and $\left|z_{2}+1-i\right|=2$, determine the maximum possible value of the expression $\left|z_{1}+z_{2}\right|$.
4. We will use the geometric interpretation of complex numbers. Namely, if $a$ and $b$ are complex numbers, then the value $|a-b|$ is equal to the distance between the complex numbers (more precisely, the points they determine) $a$ and $b$. Therefore, the set of all numbers $z_{1}$ that satisfy the condition in the problem statement represents a circle $k_{1}$ with center at $-2i$ and radius 2, while the set of all numbers $z_{2}$ that satisfy the condition in the problem statement represents a circle $k_{2}$ with center at the point $i-1$ and radius 1. Note that the value $\left|z_{1}+z_{2}\right|$ represents the distance between the complex numbers $z_{1}$ and $-z_{2}$. If we denote by $k_{2}^{\prime}$ the central-symmetric image of the circle $k_{2}$ with respect to the origin (i.e., the circle with center at the point $1-i$ and radius 1), then, as $z_{2}$ describes the circle $k_{2}$, we have that $-z_{2}$ describes the circle $k_{2}^{\prime}$, and we need the maximum distance that can be achieved between a point on $k_{1}$ and a point on $k_{2}^{\prime}$. This distance is clearly achieved for the points marked in the figure, and is equal to $2+|(1-i)-(-2i)|+1=3+|1+i|=3+\sqrt{2}$.