Find the LCM of `x^2 -4 and x^4 -16.`

Answer

To find the LCM of \(x^2 - 4\) and \(x^4 - 16\), follow these steps:

### Step 1: Factorize the given expressions

1. **Factorize \(x^2 - 4\):**
\[ x^2 - 4 = (x - 2)(x + 2) \]
This is based on the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\).

2. **Factorize \(x^4 - 16\):**
\[ x^4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) \]
Again, using the difference of squares formula.

Now, factorize \(x^2 - 4\) further:
\[ x^2 - 4 = (x - 2)(x + 2) \]

So, we have:
\[ x^4 - 16 = (x - 2)(x + 2)(x^2 + 4) \]

### Step 2: Identify the unique factors and their highest powers

From the factorizations:
- \(x^2 - 4 = (x - 2)(x + 2)\)
- \(x^4 - 16 = (x - 2)(x + 2)(x^2 + 4)\)

The unique factors are:
- \(x - 2\)
- \(x + 2\)
- \(x^2 + 4\)

### Step 3: Multiply the highest power of all unique factors

The LCM will be the product of all unique factors:
\[ \text{LCM} = (x - 2)(x + 2)(x^2 + 4) \]

### Final Answer

\[ \boxed{(x - 2)(x + 2)(x^2 + 4)} \]