LCM of the polynomials P and Q, where
`P=(x-2)(x+1)^(2)(x+3)^(2)`
`Q=(x+1)^(2)(x+3)(x+4)` is given by :

Options

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

Answer

To find the Least Common Multiple (LCM) of the polynomials \( P \) and \( Q \), we will follow these steps:

### Step 1: Write down the polynomials
Given:
- \( P = (x - 2)(x + 1)^2(x + 3)^2 \)
- \( Q = (x + 1)^2(x + 3)(x + 4) \)

### Step 2: Identify the factors of each polynomial
- The factors of \( P \) are:
- \( (x - 2) \)
- \( (x + 1)^2 \)
- \( (x + 3)^2 \)

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

### Step 3: Determine the LCM by taking the highest power of each factor
- For \( (x + 1) \):
- Highest power in \( P \) is \( (x + 1)^2 \)
- Highest power in \( Q \) is \( (x + 1)^2 \)
- Therefore, in LCM: \( (x + 1)^2 \)

- For \( (x + 3) \):
- Highest power in \( P \) is \( (x + 3)^2 \)
- Highest power in \( Q \) is \( (x + 3)^1 \)
- Therefore, in LCM: \( (x + 3)^2 \)

- For \( (x - 2) \):
- Present only in \( P \) as \( (x - 2)^1 \)
- Therefore, in LCM: \( (x - 2)^1 \)

- For \( (x + 4) \):
- Present only in \( Q \) as \( (x + 4)^1 \)
- Therefore, in LCM: \( (x + 4)^1 \)

### Step 4: Combine all the factors to get the LCM
Now, we can combine all the factors we have identified:
\[ \text{LCM}(P, Q) = (x + 1)^2 (x + 3)^2 (x - 2) (x + 4) \]

### Final Answer
Thus, the LCM of the polynomials \( P \) and \( Q \) is:
\[ (x + 1)^2 (x + 3)^2 (x - 2) (x + 4) \]

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