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.