If `2 + isqrt3` is a root of `x^(3) - 6x^(2) + px + q = 0` (where `p` and `q` are real) then `p + q` is

Explanation

To solve the problem, we need to find the values of \( p \) and \( q \) given that \( 2 + i\sqrt{3} \) is a root of the polynomial equation \( x^3 - 6x^2 + px + q = 0 \). Since the coefficients \( p \) and \( q \) are real, the complex conjugate \( 2 - i\sqrt{3} \) must also be a root.

### Step 1: Find the third root
Let the third root be \( r \). By Vieta's formulas, the sum of the roots is equal to the coefficient of \( x^2 \) with the opposite sign:
\[ (2 + i\sqrt{3}) + (2 - i\sqrt{3}) + r = 6 \]
Simplifying this, we have:
\[ 4 + r = 6 \implies r = 2 \]

### Step 2: Write the polynomial
The roots of the polynomial are \( 2 + i\sqrt{3} \), \( 2 - i\sqrt{3} \), and \( 2 \). The polynomial can be expressed as:
\[ (x - (2 + i\sqrt{3}))(x - (2 - i\sqrt{3}))(x - 2) \]

### Step 3: Simplify the first two factors
The product of the first two factors is:
\[ (x - (2 + i\sqrt{3}))(x - (2 - i\sqrt{3})) = (x - 2)^2 + 3 \]
This simplifies to:
\[ (x - 2)^2 + 3 = (x^2 - 4x + 4 + 3) = x^2 - 4x + 7 \]

### Step 4: Multiply by the third factor
Now we multiply this result by the third factor:
\[ (x^2 - 4x + 7)(x - 2) \]
Expanding this:
\[ = x^3 - 2x^2 - 4x^2 + 8x + 7x - 14 \]
Combining like terms:
\[ = x^3 - 6x^2 + 15x - 14 \]

### Step 5: Identify \( p \) and \( q \)
From the expanded polynomial \( x^3 - 6x^2 + 15x - 14 \), we can identify:
\[ p = 15 \quad \text{and} \quad q = -14 \]

### Step 6: Calculate \( p + q \)
Now we find \( p + q \):
\[ p + q = 15 + (-14) = 1 \]

### Final Answer
Thus, the value of \( p + q \) is:
\[ \boxed{1} \]