The LCM of `(x+2)^2(x-2)` and `(x^2-4x-12)` is:

Options

A. `(x+2)(x-2)^2`
B. (x-2)(x+2)(x-6)
C. (x-2)(x+2)
D. `(x-2)(x-6)(x+2)^2`

Answer

To find the LCM of the two expressions \((x+2)^2(x-2)\) and \((x^2-4x-12)\), we will follow these steps:

### Step 1: Factor the second expression

The second expression is \(x^2 - 4x - 12\). We need to factor it.

1. **Identify the coefficients**: The expression is in the form \(ax^2 + bx + c\) where \(a = 1\), \(b = -4\), and \(c = -12\).
2. **Find two numbers that multiply to \(c\) (-12) and add to \(b\) (-4)**: The numbers are -6 and 2.
3. **Rewrite the expression**:
\[ x^2 - 4x - 12 = x^2 - 6x + 2x - 12 \]
4. **Factor by grouping**:
\[ = x(x - 6) + 2(x - 6) = (x - 6)(x + 2) \]

So, we have:
\[ x^2 - 4x - 12 = (x - 6)(x + 2) \]

### Step 2: Identify the factors of both expressions

Now we have:
1. \(f(x) = (x + 2)^2(x - 2)\)
2. \(g(x) = (x - 6)(x + 2)\)

### Step 3: Determine the LCM

To find the LCM, we take the highest power of each factor present in both expressions:

1. **Factor \(x + 2\)**:
- In \(f(x)\), the power is 2.
- In \(g(x)\), the power is 1.
- The highest power is \(2\), so we take \((x + 2)^2\).

2. **Factor \(x - 2\)**:
- Present in \(f(x)\) with power 1.
- Not present in \(g(x)\).
- We take \((x - 2)^1\).

3. **Factor \(x - 6\)**:
- Not present in \(f(x)\).
- Present in \(g(x)\) with power 1.
- We take \((x - 6)^1\).

### Step 4: Write the LCM

Combining these factors, the LCM is:
\[ \text{LCM} = (x + 2)^2 (x - 2)(x - 6) \]

### Final Answer

Thus, the LCM of \((x + 2)^2(x - 2)\) and \((x^2 - 4x - 12)\) is:
\[ \text{LCM} = (x + 2)^2 (x - 2)(x - 6) \]

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