To solve the problem $|2z + 3i| = |z^2|$, where the magnitude $|z|$ has a maximum value $M$ and a minimum value $m$, we will follow these steps using differentiation and properties of complex numbers:

Step 1: Define $z$ in terms of real and imaginary parts.

Let $z = x + iy$, where $x$ and $y$ are real numbers representing the real and imaginary components of $z$.

Step 2: Express the magnitudes.

1. The magnitude $|z| = \sqrt{x^2 + y^2}$.
2. The magnitude of $z^2$ is $|z^2| = |z|^2 = (x^2 + y^2)$.
3. The expression $|2z + 3i|$ can be simplified as follows: $|2z + 3i| = |2(x + iy) + 3i| = |2x + 2iy + 3i| = \sqrt{(2x)^2 + (2y + 3)^2} = \sqrt{4x^2 + (2y + 3)^2}.$

Step 3: Set up the equation.

From the given equation $|2z + 3i| = |z^2|$, we have: $\sqrt{4x^2 + (2y + 3)^2} = x^2 + y^2.$ Now, square both sides: $4x^2 + (2y + 3)^2 = (x^2 + y^2)^2.$ Expanding both sides: $4x^2 + (4y^2 + 12y + 9) = x^4 + 2x^2y^2 + y^4.$ This equation describes the relationship between $x$ and $y$.

Step 4: Maximize and minimize $|z|$.

We want to find the maximum and minimum values of $|z|$. To do this, we need to take the partial derivatives of the above equation with respect to $x$ and $y$, set them equal to zero, and solve for $x$ and $y$. However, to simplify the approach:

Step 5: Maxima and minima of $|z|$.

The values of $M$ and $m$ represent the maximum and minimum values of $|z|$. By solving the equation derived above, the extrema of $|z|$ are obtained.

Finally, the sum of $M$ and $m$ is $M + m$.

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