Let `z_(1)` and `z_(2)` be complex numbers such that `z_(1)^(2)-4z_(2)=16+20i` and the roots `alpha` and `beta` of `x^(2)+z_(1)x+z_(2)+m=0` for some complex number `m` satisfies `|alpha-beta|=2sqrt(7)`.
The maximum value of `|m|` is

Options

A. 14
B. `2 sqrt7`
C. `7 + sqrt41`
D. `28 + sqrt(41)`

Answer

To solve the problem step by step, we will start with the given equations and conditions.

### Step 1: Understand the Given Information
We have two complex numbers \( z_1 \) and \( z_2 \) such that:
\[ z_1^2 - 4z_2 = 16 + 20i \]
We also know that the roots \( \alpha \) and \( \beta \) of the quadratic equation:
\[ x^2 + z_1 x + (z_2 + m) = 0 \]
satisfy the condition:
\[ |\alpha - \beta| = 2\sqrt{7} \]

### Step 2: Relate Roots to Coefficients
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -z_1 \)
- The product of the roots \( \alpha \beta = z_2 + m \)

### Step 3: Find the Expression for \( |\alpha - \beta| \)
The expression for the difference of the roots is given by:
\[ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta} \]
Substituting the values from Vieta's formulas:
\[ |\alpha - \beta| = \sqrt{(-z_1)^2 - 4(z_2 + m)} \]
We know that \( |\alpha - \beta| = 2\sqrt{7} \), so we can set up the equation:
\[ 2\sqrt{7} = \sqrt{z_1^2 - 4(z_2 + m)} \]
Squaring both sides gives:
\[ 4 \cdot 7 = z_1^2 - 4(z_2 + m) \]
This simplifies to:
\[ 28 = z_1^2 - 4z_2 - 4m \]

### Step 4: Substitute \( z_1^2 - 4z_2 \)
From the first equation, we know:
\[ z_1^2 - 4z_2 = 16 + 20i \]
Substituting this into our equation gives:
\[ 28 = (16 + 20i) - 4m \]
Rearranging this, we find:
\[ 4m = (16 + 20i) - 28 \]
\[ 4m = -12 + 20i \]
Dividing by 4:
\[ m = -3 + 5i \]

### Step 5: Find the Maximum Value of \( |m| \)
To find the maximum value of \( |m| \):
\[ |m| = |-3 + 5i| = \sqrt{(-3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

### Step 6: Analyze the Circle Condition
We need to maximize \( |m| \) under the condition derived from the equation of a circle:
\[ |m - (4 + 5i)| = 7 \]
The center of the circle is at \( (4, 5) \) with a radius of \( 7 \). The maximum distance from the origin (0, 0) to any point on the circle occurs when the line from the origin to the center of the circle is extended by the radius.

### Step 7: Calculate the Maximum Distance
The distance from the origin to the center \( (4, 5) \) is:
\[ \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]
Thus, the maximum value of \( |m| \) is:
\[ \sqrt{41} + 7 \]

### Final Answer
The maximum value of \( |m| \) is:
\[ \boxed{7 + \sqrt{41}} \]