If `veca` and `vecb` are non-zero, non parallel vectors, then the value of
`|veca + vecb+veca xx vecb|^(2) +|veca + vecb-veca xx vecb|^(2)` equals

Options

A. `(1+veca.veca)(1+vecb.vecc)`
B. `2(1+veca.veca)(1+vecb.vecb)`
C. `2{(1+veca.veca)(1+vecb.vecb)-(1-veca.vecb)^(2)}`
D. `2{(1-veca.veca)(1-vecb.vecb)+(1-veca.vecb)^(2)}`

Explanation

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

\[ | \vec{a} + \vec{b} + \vec{a} \times \vec{b} |^2 + | \vec{a} + \vec{b} - \vec{a} \times \vec{b} |^2 \]

### Step 1: Expand the Magnitudes
We start by expanding the magnitudes of both expressions.

1. **First Expression:**
\[ | \vec{a} + \vec{b} + \vec{a} \times \vec{b} |^2 = (\vec{a} + \vec{b} + \vec{a} \times \vec{b}) \cdot (\vec{a} + \vec{b} + \vec{a} \times \vec{b}) \]
Using the distributive property:
\[ = \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} + (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{b}) + 2(\vec{a} \cdot \vec{b}) + 2(\vec{a} + \vec{b}) \cdot (\vec{a} \times \vec{b}) \]

2. **Second Expression:**
\[ | \vec{a} + \vec{b} - \vec{a} \times \vec{b} |^2 = (\vec{a} + \vec{b} - \vec{a} \times \vec{b}) \cdot (\vec{a} + \vec{b} - \vec{a} \times \vec{b}) \]
Similarly, expanding this:
\[ = \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} + (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{b}) + 2(\vec{a} \cdot \vec{b}) - 2(\vec{a} + \vec{b}) \cdot (\vec{a} \times \vec{b}) \]

### Step 2: Combine the Two Expressions
Now we combine the two expanded expressions:

\[ | \vec{a} + \vec{b} + \vec{a} \times \vec{b} |^2 + | \vec{a} + \vec{b} - \vec{a} \times \vec{b} |^2 \]

This results in:
\[ 2(\vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b}) + 2(\vec{a} \cdot \vec{b}) + 2(\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{b}) \]

### Step 3: Simplify the Terms
Recall that:
- \(\vec{a} \cdot \vec{a} = |\vec{a}|^2\)
- \(\vec{b} \cdot \vec{b} = |\vec{b}|^2\)
- \((\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{b}) = |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \sin^2 \theta\)

Thus, we can rewrite:
\[ = 2(|\vec{a}|^2 + |\vec{b}|^2 + |\vec{a}|^2 |\vec{b}|^2 \sin^2 \theta + 2(\vec{a} \cdot \vec{b})) \]

### Step 4: Final Expression
Now we can express the final result in terms of the dot products and magnitudes:
\[ = 2(1 + |\vec{a}|^2)(1 + |\vec{b}|^2) - 2(1 - \vec{a} \cdot \vec{b})^2 \]

### Conclusion
Thus, the final answer is:
\[ 2(1 + |\vec{a}|^2)(1 + |\vec{b}|^2) - 2(1 - \vec{a} \cdot \vec{b})^2 \]