Let ` z_(1) = 3 + 4i and z_(2) = - 1 + 2i " then " | z_(1) + z_(2)|^(2) - 2 (| z_(1)|^(2) + | z_(2)|^(2))` is equal to

Options

A. `| z_(1) - z_(2)|^(2)`
B. ` - | z_(1) - z_(2)|^(2)`
C. `| z_(1) + z_(2)|^(2)`
D. none of these

Answer

To solve the problem, we need to evaluate the expression:

\[ | z_1 + z_2 |^2 - 2 (| z_1 |^2 + | z_2 |^2) \]

Given:
- \( z_1 = 3 + 4i \)
- \( z_2 = -1 + 2i \)

### Step 1: Calculate \( z_1 + z_2 \)

First, we add the two complex numbers:

\[ z_1 + z_2 = (3 + 4i) + (-1 + 2i) = (3 - 1) + (4 + 2)i = 2 + 6i \]

### Step 2: Calculate \( | z_1 + z_2 |^2 \)

Next, we find the modulus of \( z_1 + z_2 \):

\[ | z_1 + z_2 | = | 2 + 6i | = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \]

Now, we square this modulus:

\[ | z_1 + z_2 |^2 = (2\sqrt{10})^2 = 4 \cdot 10 = 40 \]

### Step 3: Calculate \( | z_1 |^2 \) and \( | z_2 |^2 \)

Now, we calculate the moduli of \( z_1 \) and \( z_2 \):

\[ | z_1 | = | 3 + 4i | = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
\[ | z_1 |^2 = 5^2 = 25 \]

\[ | z_2 | = | -1 + 2i | = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]
\[ | z_2 |^2 = (\sqrt{5})^2 = 5 \]

### Step 4: Calculate \( | z_1 |^2 + | z_2 |^2 \)

Now, we add the squares of the moduli:

\[ | z_1 |^2 + | z_2 |^2 = 25 + 5 = 30 \]

### Step 5: Substitute into the expression

Now we substitute everything back into the original expression:

\[ | z_1 + z_2 |^2 - 2 (| z_1 |^2 + | z_2 |^2) = 40 - 2 \cdot 30 = 40 - 60 = -20 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{-20} \]