The value of ` (z + 3) (barz + 3) ` is equivlent to (A) |z+3|^(2) (B) |z-3| (C) z^2+3 (D) none of these

Options

A. `| z+ 3|^(2)`
B. `|z -3|`
C. `z^(2) +3`
D. None of these

Explanation

To solve the expression \( (z + 3)(\bar{z} + 3) \), we can start by substituting \( z \) with its rectangular form. Let's denote \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of \( z \). The conjugate of \( z \), denoted as \( \bar{z} \), is \( x - iy \).

### Step 1: Substitute \( z \) and \( \bar{z} \)
We have:
\[ z + 3 = (x + iy) + 3 = (x + 3) + iy \]
\[ \bar{z} + 3 = (x - iy) + 3 = (x + 3) - iy \]

### Step 2: Write the expression
Now, we can write the expression:
\[ (z + 3)(\bar{z} + 3) = ((x + 3) + iy)((x + 3) - iy) \]

### Step 3: Use the difference of squares
This expression is in the form of \( (a + b)(a - b) \), which simplifies to \( a^2 - b^2 \):
\[ = (x + 3)^2 - (iy)^2 \]

### Step 4: Simplify the expression
Now, we simplify \( (iy)^2 \):
\[ (iy)^2 = i^2y^2 = -y^2 \]
Thus, we have:
\[ = (x + 3)^2 - (-y^2) = (x + 3)^2 + y^2 \]

### Step 5: Recognize the modulus
The expression \( (x + 3)^2 + y^2 \) is the square of the modulus of the complex number \( z + 3 \):
\[ = |z + 3|^2 \]

### Conclusion
Thus, the value of \( (z + 3)(\bar{z} + 3) \) is equivalent to \( |z + 3|^2 \).

### Final Answer
The correct option is (A) \( |z + 3|^2 \).

---