The equation `z^2 - (3 + i) z + (m + 2i) = 0 m in R` , has exactly one real and one non real complex root, then product of real root and imaginary part of non-real complex root is:

Explanation

To solve the problem, we need to analyze the quadratic equation given:

\[ z^2 - (3 + i)z + (m + 2i) = 0 \]

where \( m \in \mathbb{R} \). We want to find the product of the real root and the imaginary part of the non-real complex root when the equation has exactly one real root and one non-real complex root.

### Step 1: Identify the roots
Let the real root be \( \alpha \). Since the equation has one real root and one non-real complex root, we can express the roots using Vieta's formulas. The sum of the roots is given by:

\[ \alpha + z_1 = 3 + i \]

and the product of the roots is:

\[ \alpha z_1 = m + 2i \]

### Step 2: Separate real and imaginary parts
From the sum of the roots, we can separate the real and imaginary parts:

1. Real part: \( \alpha + \text{Re}(z_1) = 3 \)
2. Imaginary part: \( \text{Im}(z_1) = 1 \)

From the imaginary part, we can deduce that:

\[ \text{Im}(z_1) = 1 \]

### Step 3: Solve for \( \alpha \)
Now, substituting \( \text{Im}(z_1) = 1 \) into the equation for the real part:

\[ \alpha + \text{Re}(z_1) = 3 \]

Let \( \text{Re}(z_1) = x \). Then we have:

\[ \alpha + x = 3 \quad \text{(1)} \]

### Step 4: Substitute into the product of the roots
Next, we use the product of the roots:

\[ \alpha z_1 = m + 2i \]

Substituting \( z_1 = x + i \):

\[ \alpha (x + i) = m + 2i \]

Separating real and imaginary parts gives us:

1. Real part: \( \alpha x = m \)
2. Imaginary part: \( \alpha = 2 \)

### Step 5: Solve for \( m \)
From the imaginary part, we have:

\[ \alpha = 2 \]

Substituting \( \alpha = 2 \) into equation (1):

\[ 2 + x = 3 \implies x = 1 \]

Now substituting \( \alpha = 2 \) and \( x = 1 \) into the equation for \( m \):

\[ m = \alpha x = 2 \cdot 1 = 2 \]

### Step 6: Identify the roots
Now we have \( \alpha = 2 \) and \( z_1 = 1 + i \).

### Step 7: Calculate the product
The imaginary part of the non-real complex root \( z_1 \) is \( \text{Im}(z_1) = 1 \).

Now, we find the product of the real root \( \alpha \) and the imaginary part of the non-real complex root:

\[ \text{Product} = \alpha \cdot \text{Im}(z_1) = 2 \cdot 1 = 2 \]

### Final Answer
Thus, the product of the real root and the imaginary part of the non-real complex root is:

\[ \boxed{2} \]