Simplify this expression: 8((2-x)^2(2+x)^2-x((2x-4)(2+x)^2+(2x+4)(2-x^2)))


Let's simplify the given expression step by step.

The expression is:

\[
8\left( (2-x)^2(2+x)^2 - x\left( (2x-4)(2+x)^2 + (2x+4)(2-x)^2 \right) \right)
\]

### Step 1: Expand the squares and products

1. **Expand \((2-x)^2\) and \((2+x)^2\):**

   \[
   (2-x)^2 = 4 - 4x + x^2
   \]
   \[
   (2+x)^2 = 4 + 4x + x^2
   \]

2. **Now, multiply these two expressions:**

   \[
   (4 - 4x + x^2)(4 + 4x + x^2)
   \]

   Use the distributive property (FOIL method):

   \[
   = 16 + 16x - 16x - 16x^2 + 4x^2 + 4x^2 - 16x^2 + x^4
   \]

   Combine like terms:

   \[
   = x^4 - 12x^2 + 16
   \]

3. **Now, expand the terms inside the expression with \(x\):**

   - For \((2x-4)(2+x)^2\):

     \[
     = (2(x-2))(4 + 4x + x^2) = 2(4x - 8 + 4x^2 - 16x + 2x^3 - 8x^2) = 2(2x^3 - 4x^2 - 12x + 16)
     \]

   - For \((2x+4)(2-x)^2\):

     \[
     = (2(x+2))(4 - 4x + x^2) = 2(4x + 8 - 4x^2 - 8x + 2x^3 - 8x^2) = 2(2x^3 - 12x^2 + 4x + 16)
     \]

4. **Now, combine the expanded terms:**

   Subtract \(x\) times these expanded terms from the first part:

   \[
   x((2x-4)(2+x)^2 + (2x+4)(2-x)^2) 
   = x\left(2(2x^3 - 4x^2 - 12x + 16) + 2(2x^3 - 12x^2 + 4x + 16)\right)
   \]

5. **Simplify this whole expression:**

   Simplify it by multiplying and then substituting back into the expression:

   \[
   = 8\left( (x^4 - 12x^2 + 16) - 4x^4 + 32x^3 - 64x^2 + 64x \right)
   \]

6. **Further simplify and combine like terms:**

   After combining and simplifying:

   \[
   = 8\left( -3x^4 + 32x^3 + 64x - 48 \right)
   \]

7. **Final answer:**

   Simplifying the result:

   \[
   -24x^4 + 256x^3 + 512x - 384
   \]

This is the simplified form of the expression.

Would you like to explore any specific step or details? 

### Related Questions:
1. How do you expand binomials using the FOIL method?
2. How can you simplify polynomial expressions?
3. What are common mistakes when simplifying expressions?
4. How do you combine like terms in algebraic expressions?
5. What is the distributive property, and how is it applied in algebra?

**Tip:** Always double-check each step when simplifying complex expressions to avoid small errors that can affect the final answer.

Learn how to simplify the complex algebraic expression 8((2-x)^2(2+x)^2-x((2x-4)(2+x)^2+(2x+4)(2-x)^2)). This problem involves expanding binomials using the FOIL method, simplifying polynomial expressions, and combining like terms. Suitable for advanced high school students.

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