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The question asks for the least common multiple (LCM) of the polynomials \(x^2 - 16\) and \(x^2 + 4x - 32\). Here’s how we solve it:

### Step 1: Factorize the polynomials
1. **Factorize \(x^2 - 16\):**
   \[
   x^2 - 16 = (x - 4)(x + 4)
   \]

2. **Factorize \(x^2 + 4x - 32\):**
   To factorize \(x^2 + 4x - 32\), find two numbers whose product is \(-32\) and sum is \(4\). These numbers are \(8\) and \(-4\):
   \[
   x^2 + 4x - 32 = (x + 8)(x - 4)
   \]

### Step 2: Find the LCM
The LCM of two polynomials includes all unique factors raised to their highest powers:
- From \(x^2 - 16\): \((x - 4)\) and \((x + 4)\)
- From \(x^2 + 4x - 32\): \((x + 8)\) and \((x - 4)\)

The LCM is:
\[
\text{LCM} = (x - 4)(x + 4)(x + 8)
\]

### Final Answer:
The least common multiple is:
\[
(x - 4)(x + 4)(x + 8)
\]

Would you like more details or a breakdown of specific steps? Here are some related questions to expand your understanding:

1. How do you verify that a polynomial is factored correctly?
2. What are the differences between LCM and GCD for polynomials?
3. How do you handle higher-degree polynomials when finding the LCM?
4. Can the LCM of polynomials be simplified further if specific values are given?
5. How do these concepts apply to real-world problems involving algebra?

**Tip:** Always double-check your factorization to avoid errors in finding the LCM.

What is the least common multiple of x^2 - 16 and x^2 + 4x - 32?

Discover how to calculate the least common multiple (LCM) of two polynomials: x^2 - 16 and x^2 + 4x - 32. This step-by-step guide explores factoring, identifying unique factors, and applying the distributive property to solve the problem. Perfect for Grades 9-11 students learning advanced algebra concepts.

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