The LCM of the polynomials `(x + 3) ^(2) (x - 2) ( x +1) ^(2) and (x +1) ^(3) (x + 3) (x +4)` is

Options

A. `(x -2) (x + 1)^(3) (x +4)`
B. `(x -2) (x +1) ^(3) (x + 3) (x +A)`
C. `(x -2) (x + 3) (x + 4)`
D. `(x - 2) ^(2) (x + 1) (x + 3) ^(2) (x + 4)`

Answer

To find the LCM of the polynomials \( (x + 3)^{2} (x - 2) (x + 1)^{2} \) and \( (x + 1)^{3} (x + 3) (x + 4) \), we can follow these steps:

### Step 1: Factor the polynomials
We start by identifying the factors of each polynomial:

1. **First Polynomial:**
\[ (x + 3)^{2} (x - 2) (x + 1)^{2} \]
- Factors: \( (x + 3)^{2}, (x - 2), (x + 1)^{2} \)

2. **Second Polynomial:**
\[ (x + 1)^{3} (x + 3) (x + 4) \]
- Factors: \( (x + 1)^{3}, (x + 3), (x + 4) \)

### Step 2: Identify the highest powers of each factor
Next, we need to find the highest power of each factor that appears in either polynomial:

- **For \( (x + 1) \):**
- First Polynomial: \( (x + 1)^{2} \)
- Second Polynomial: \( (x + 1)^{3} \)
- Highest Power: \( (x + 1)^{3} \)

- **For \( (x + 3) \):**
- First Polynomial: \( (x + 3)^{2} \)
- Second Polynomial: \( (x + 3)^{1} \)
- Highest Power: \( (x + 3)^{2} \)

- **For \( (x - 2) \):**
- First Polynomial: \( (x - 2)^{1} \)
- Second Polynomial: Not present
- Highest Power: \( (x - 2)^{1} \)

- **For \( (x + 4) \):**
- First Polynomial: Not present
- Second Polynomial: \( (x + 4)^{1} \)
- Highest Power: \( (x + 4)^{1} \)

### Step 3: Write the LCM
Now that we have identified the highest powers of each factor, we can write the LCM:

\[ \text{LCM} = (x + 1)^{3} (x + 3)^{2} (x - 2)^{1} (x + 4)^{1} \]

### Step 4: Simplify the expression
The final expression for the LCM can be written as:

\[ \text{LCM} = (x + 1)^{3} (x + 3)^{2} (x - 2) (x + 4) \]

### Conclusion
Thus, the LCM of the given polynomials is:

\[ (x + 1)^{3} (x + 3)^{2} (x - 2) (x + 4) \]

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