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To solve this problem, we need to find the result of the cross product \((\vec{u} - 4\vec{v}) \times (\vec{u} + 2\vec{v})\), given that \(\vec{u} \times \vec{v} = (2, -4, 3)\).

Let's break down the solution step by step:

### Step 1: Expanding the Cross Product
The expression \((\vec{u} - 4\vec{v}) \times (\vec{u} + 2\vec{v})\) can be expanded using the distributive property of the cross product:
\[
(\vec{u} - 4\vec{v}) \times (\vec{u} + 2\vec{v}) = \vec{u} \times \vec{u} + \vec{u} \times (2\vec{v}) - (4\vec{v}) \times \vec{u} - (4\vec{v}) \times (2\vec{v})
\]

### Step 2: Simplifying Each Term
1. **\(\vec{u} \times \vec{u} = \vec{0}\)**: The cross product of a vector with itself is always zero.

2. **\(\vec{u} \times (2\vec{v})\)**: By the scalar multiple rule, this becomes:
   \[
   \vec{u} \times (2\vec{v}) = 2(\vec{u} \times \vec{v}) = 2 \cdot (2, -4, 3) = (4, -8, 6)
   \]

3. **\((-4\vec{v}) \times \vec{u}\)**: We can use the anti-commutative property of the cross product, \(\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})\), to rewrite this term as:
   \[
   (-4\vec{v}) \times \vec{u} = -4 (\vec{v} \times \vec{u}) = -4 (-(\vec{u} \times \vec{v})) = 4 (\vec{u} \times \vec{v}) = 4 \cdot (2, -4, 3) = (8, -16, 12)
   \]

4. **\((-4\vec{v}) \times (2\vec{v}) = \vec{0}\)**: The cross product of two parallel vectors (scalar multiples of each other) is zero.

### Step 3: Adding the Results
Now, we add the non-zero results from each term:
\[
(4, -8, 6) + (8, -16, 12) = (12, -24, 18)
\]

### Final Answer
The result of \((\vec{u} - 4\vec{v}) \times (\vec{u} + 2\vec{v})\) is:
\[
(12, -24, 18)
\]

If you have any questions or would like further details, feel free to ask!

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### Related Questions
1. How is the cross product of two vectors calculated in general?
2. Why does the cross product of a vector with itself always result in zero?
3. How does the distributive property of the cross product work?
4. What is the geometric interpretation of the cross product in three dimensions?
5. How do scalar multiples affect the cross product?

### Tip
Always remember that the cross product of parallel vectors is zero, which can simplify many calculations.

Given \( \vec{u} \times \vec{v} = (2, -4, 3) \), find \((\vec{u} - 4\vec{v}) \times (\vec{u} + 2\vec{v})\).

Learn how to find the cross product \((\vec{u} - 4\vec{v}) \times (\vec{u} + 2\vec{v})\) given \( \vec{u} \times \vec{v} = (2, -4, 3) \). This solution includes vector algebra concepts like the distributive property and the anti-commutative property of cross products, ideal for high school students in Grades 11-12.

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