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 \( m \) is a real number. The goal is to find the product of the real root and the imaginary part of the non-real complex root.

### Step 1: Identify the nature of the roots
For the quadratic equation to have exactly one real root and one non-real complex root, the discriminant must be less than zero. The discriminant \( D \) of the equation \( az^2 + bz + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

Here, \( a = 1 \), \( b = -(3 + i) \), and \( c = m + 2i \). Thus, we compute:

\[ D = (-(3 + i))^2 - 4(1)(m + 2i) \]

### Step 2: Calculate the discriminant
Calculating \( (-(3 + i))^2 \):

\[ D = (3 + i)^2 - 4(m + 2i) \]
\[ = 9 + 6i - 1 - 4m - 8i \]
\[ = 8 - 4m - 2i \]

### Step 3: Set the condition for the discriminant
For the roots to be one real and one non-real, we need:

\[ D < 0 \]

This means both the real part and the imaginary part of \( D \) must satisfy:

1. Real part: \( 8 - 4m < 0 \)
2. Imaginary part: \( -2 < 0 \) (which is always true)

From the first inequality:

\[ 8 < 4m \implies m > 2 \]

### Step 4: Find the real root
Now, we need to find the real root. Let’s denote the real root as \( \alpha \). From Vieta's formulas, the sum of the roots \( \alpha + z_1 = 3 + i \).

### Step 5: Substitute the real root into the equation
We can substitute \( \alpha \) into the equation:

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

### Step 6: Separate real and imaginary parts
Separating the real and imaginary parts gives us:

1. Real part: \( \alpha^2 - 3\alpha + m = 0 \)
2. Imaginary part: \( -\alpha + 2 = 0 \)

From the imaginary part equation:

\[ \alpha = 2 \]

### Step 7: Substitute back to find \( m \)
Substituting \( \alpha = 2 \) into the real part equation:

\[ 2^2 - 3(2) + m = 0 \]
\[ 4 - 6 + m = 0 \implies m = 2 \]

### Step 8: Find the complex root
Now, substituting \( m = 2 \) back into the original equation:

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

Using Vieta's formulas again, the complex root \( z_1 \) can be calculated:

\[ 2 + z_1 = 3 + i \implies z_1 = 1 + i \]

### Step 9: Find the imaginary part of the complex root
The imaginary part of the complex root \( z_1 = 1 + i \) is \( 1 \).

### Step 10: Calculate the product
Finally, we calculate the product of the real root \( \alpha \) and the imaginary part of the complex root:

\[ \text{Product} = \alpha \cdot \text{(Imaginary part of } z_1) = 2 \cdot 1 = 2 \]

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

\[ \boxed{2} \]