### Instruction: Let $z_1,$ $z_2,$ $z_3$ be complex numbers such that $|z_1| = 2,$ $|z_2| = 3,$ and $|z_3| = 4.$  Find the largest possible value of
\[|z_1 - z_2|^2 + |z_1 - z_3|^2 + |z_2 - z_3|^2.\].### Response: I notice that this expression is the sum of the squares of the lengths of the sides of a triangle with vertices at $z_1,$ $z_2,$ and $z_3.$
To maximize this expression, I want to make the triangle as large as possible.
I recall that the triangle inequality says that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides.
So, the largest possible value of $|z_1 - z_2|$ is $|z_1| + |z_2| = 2 + 3 = 5.$
Similarly, the largest possible values of $|z_1 - z_3|$ and $|z_2 - z_3|$ are $|z_1| + |z_3| = 2 + 4 = 6$ and $|z_2| + |z_3| = 3 + 4 = 7,$ respectively.
Therefore, the largest possible value of the expression is $5^2 + 6^2 + 7^2 = 25 + 36 + 49 = 110.$