Are you sure this: =16+16*x*−16*x*−16*x*2+4*x*2+4*x*2−16*x*2+*x*4, can be simplified to this: *x*4−12*x*2+16?

You are correct to question that step; the simplification provided was incorrect. Let's carefully re-expand and simplify the expression.

### Step-by-Step Correction

We need to expand and simplify:

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

### Step 1: Expand each binomial

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

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

2. **Expand \((2+x)^2\):**

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

### Step 2: Multiply the two expanded expressions

Now, multiply \((4 - 4x + x^2)\) by \((4 + 4x + x^2)\):

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

### Step 3: Use the distributive property (FOIL method)

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

Expanding each term:

\[
= 16 + 16x + 4x^2 - 16x - 16x^2 - 4x^3 + 4x^2 + 4x^3 + x^4
\]

### Step 4: Combine like terms

Let's combine the like terms:

- The \(x^3\) terms: \(-4x^3 + 4x^3 = 0\) (these cancel each other out)
- The \(x^2\) terms: \(4x^2 - 16x^2 + 4x^2 = -8x^2\)
- The linear \(x\) terms: \(16x - 16x = 0\) (these cancel each other out)
- The constant term: \(16\)
- The \(x^4\) term: \(x^4\)

Thus, the correct expanded form is:

\[
x^4 - 8x^2 + 16
\]

### Conclusion

So the correct simplification of the product of the binomials \((2-x)^2(2+x)^2\) is:

\[
x^4 - 8x^2 + 16
\]

Thank you for pointing that out! 

Would you like to proceed with the remaining steps of the original expression?

### Related Questions:
1. What is the correct method for expanding binomials squared?
2. How do you simplify expressions involving polynomial multiplication?
3. How can you recognize and cancel out terms in polynomial expressions?
4. Why do certain terms cancel out when multiplying polynomials?
5. What are common pitfalls in simplifying algebraic expressions?

**Tip:** Carefully track each term when performing algebraic expansions to ensure all like terms are correctly combined.

Learn how to correctly expand and simplify the product of binomials (2-x)^2(2+x)^2 with a step-by-step correction. This problem covers key algebraic concepts, including binomial expansion and polynomial simplification. Suitable for students in Grades 10-12.

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