The LCM of the polynomials A and B, where `A = (x+3)^(2)(x-2)(x+1)^(2)` and
`B = (x+1)^(2)(x+3)(x+4)`, is given by

Options

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

Answer

To find the LCM of the polynomials \( A \) and \( B \), we will follow these steps:

### Step 1: Identify the polynomials
We have:
- \( A = (x + 3)^2 (x - 2) (x + 1)^2 \)
- \( B = (x + 1)^2 (x + 3) (x + 4) \)

### Step 2: Factor the polynomials
The factors of \( A \) are:
- \( (x + 3)^2 \)
- \( (x - 2) \)
- \( (x + 1)^2 \)

The factors of \( B \) are:
- \( (x + 1)^2 \)
- \( (x + 3) \)
- \( (x + 4) \)

### Step 3: Determine the highest power of each factor
Next, we identify the highest power of each factor present in both polynomials:

- For \( (x + 3) \):
- In \( A \), the power is \( 2 \).
- In \( B \), the power is \( 1 \).
- **Take the maximum**: \( (x + 3)^2 \)

- For \( (x - 2) \):
- In \( A \), the power is \( 1 \).
- In \( B \), it is not present (power \( 0 \)).
- **Take the maximum**: \( (x - 2)^1 \)

- For \( (x + 1) \):
- In \( A \), the power is \( 2 \).
- In \( B \), the power is \( 2 \).
- **Take the maximum**: \( (x + 1)^2 \)

- For \( (x + 4) \):
- In \( A \), it is not present (power \( 0 \)).
- In \( B \), the power is \( 1 \).
- **Take the maximum**: \( (x + 4)^1 \)

### Step 4: Write the LCM
Now we combine all the factors with their highest powers to write the LCM:
\[ \text{LCM}(A, B) = (x + 3)^2 (x - 2) (x + 1)^2 (x + 4) \]

### Step 5: Final expression
Thus, the LCM of the polynomials \( A \) and \( B \) is:
\[ \text{LCM}(A, B) = (x + 3)^2 (x - 2) (x + 1)^2 (x + 4) \]