Let `theta` be the angle between the vectors `veca and vecb` , where `|veca| = 4 , |vecb| =3 ` and `theta in (pi/4, pi/3)` Then `|(veca-vecb)xx(veca+vecb)|^2+4(veca.vecb)^2` is equal to ______

Answer

To solve the problem, we need to evaluate the expression \( |(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})|^2 + 4(\vec{a} \cdot \vec{b})^2 \).

### Step 1: Calculate the cross product
We start by using the property of the cross product:
\[ |(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})|^2 = |(\vec{a} \times \vec{a}) + (\vec{a} \times \vec{b}) - (\vec{b} \times \vec{a}) - (\vec{b} \times \vec{b})|^2 \]
Since the cross product of any vector with itself is zero, we have:
\[ |(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})|^2 = |0 + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} + 0|^2 = |2(\vec{a} \times \vec{b})|^2 \]
Thus,
\[ |(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})|^2 = 4|\vec{a} \times \vec{b}|^2 \]

### Step 2: Calculate the dot product
Next, we need to calculate \( 4(\vec{a} \cdot \vec{b})^2 \). The dot product can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]
Given that \( |\vec{a}| = 4 \) and \( |\vec{b}| = 3 \), we have:
\[ \vec{a} \cdot \vec{b} = 4 \cdot 3 \cdot \cos(\theta) = 12 \cos(\theta) \]
Thus,
\[ 4(\vec{a} \cdot \vec{b})^2 = 4(12 \cos(\theta))^2 = 576 \cos^2(\theta) \]

### Step 3: Combine the results
Now we combine the results from Step 1 and Step 2:
\[ |(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})|^2 + 4(\vec{a} \cdot \vec{b})^2 = 4|\vec{a} \times \vec{b}|^2 + 576 \cos^2(\theta) \]

### Step 4: Calculate \( |\vec{a} \times \vec{b}|^2 \)
The magnitude of the cross product can be calculated as:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta) = 4 \cdot 3 \cdot \sin(\theta) = 12 \sin(\theta) \]
Thus,
\[ |\vec{a} \times \vec{b}|^2 = (12 \sin(\theta))^2 = 144 \sin^2(\theta) \]
Now substituting this back, we have:
\[ 4|\vec{a} \times \vec{b}|^2 = 4 \cdot 144 \sin^2(\theta) = 576 \sin^2(\theta) \]

### Step 5: Final Expression
Now we can combine everything:
\[ 576 \sin^2(\theta) + 576 \cos^2(\theta) = 576(\sin^2(\theta) + \cos^2(\theta)) = 576 \cdot 1 = 576 \]

### Final Answer
Thus, the final value is:
\[ \boxed{576} \]