If $$ (3 + 2\sqrt{2}) $$ is a root of the equation $$ ax^2 + bx + c = 0 $$, and $$ (4 + 2\sqrt{3}) $$ is a root of the equation $$ ay^2 + my + n = 0 $$, where $$ a, b, c, m, $$ and $$ n $$ are integers, then the value of $$ \frac{b}{m} + \frac{c - 2b}{n} $$ is:

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#### Step 1: Analyze the First Quadratic Equation  
The first quadratic equation is $$ ax^2 + bx + c = 0 $$, with a root $$ (3 + 2\sqrt{2}) $$.  
- The other root must be $$ (3 - 2\sqrt{2}) $$, as the coefficients are integers.

Using the sum and product of roots:  
- **Sum of roots**:  
  $$ (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6. $$  
  Hence,  
  $$ -\frac{b}{a} = 6 \implies b = -6a. $$

- **Product of roots**:  
  $$ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 9 - (2\sqrt{2})^2 = 9 - 8 = 1. $$  
  Hence,  
  $$ \frac{c}{a} = 1 \implies c = a. $$

#### Step 2: Analyze the Second Quadratic Equation  
The second quadratic equation is $$ ay^2 + my + n = 0 $$, with a root $$ (4 + 2\sqrt{3}) $$.  
- The other root must be $$ (4 - 2\sqrt{3}) $$.

Using the sum and product of roots:  
- **Sum of roots**:  
  $$ (4 + 2\sqrt{3}) + (4 - 2\sqrt{3}) = 8. $$  
  Hence,  
  $$ -\frac{m}{a} = 8 \implies m = -8a. $$

- **Product of roots**:  
  $$ (4 + 2\sqrt{3})(4 - 2\sqrt{3}) = 16 - (2\sqrt{3})^2 = 16 - 12 = 4. $$  
  Hence,  
  $$ \frac{n}{a} = 4 \implies n = 4a. $$

#### Step 3: Substitute Values into the Expression
We need to calculate:
$$
\frac{b}{m} + \frac{c - 2b}{n}.
$$
Substitute $$ b = -6a, c = a, m = -8a, n = 4a $$:

1. **Calculate $$ b/m $$:**  
   $$
   b/m = (-6a)/(-8a) = \frac{6}{8} = \frac{3}{4}.
   $$

2. **Calculate $$ c - 2b/n $$:**  
   $$
   c - 2b = a - [2(-6a)] = a + 12a = 13a.
   $$
   Divide by $$ n = 4a $$:  
   $$
   (c - 2b)/n = (13a)/(4a) = \frac{13}{4}.
   $$

3. **Combine Terms:**  
   $$
   \frac{b}{m} + \frac{c - 2b}{n} = \frac{3}{4} + \frac{13}{4} = \frac{16}{4} = 4.
   $$

Thus, the value is **4**, and the correct answer is **C**.

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