Find the LCM and HCF of the polynomials `P(x)= (x+1)^(2) (x+2)`
`Q(x)= (x+1) (x-2)`

Answer

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the polynomials \( P(x) = (x+1)^2 (x+2) \) and \( Q(x) = (x+1)(x-2) \), we will follow these steps:

### Step 1: Factor the Polynomials
First, we will write down the factored forms of both polynomials.

- \( P(x) = (x+1)^2 (x+2) \)
- \( Q(x) = (x+1)(x-2) \)

### Step 2: Identify the Common and Unique Factors
Next, we identify the factors in both polynomials.

- The factors of \( P(x) \) are \( (x+1)^2 \) and \( (x+2) \).
- The factors of \( Q(x) \) are \( (x+1) \) and \( (x-2) \).

### Step 3: Find the HCF
The HCF is found by taking the common factors with the lowest power.

- The common factor is \( (x+1) \).
- The lowest power of \( (x+1) \) in both polynomials is \( (x+1)^1 \).

Thus, the HCF is:
\[ \text{HCF} = (x+1) \]

### Step 4: Find the LCM
The LCM is found by taking all the factors, with the highest power for the common factors.

- For \( (x+1) \), the highest power is \( (x+1)^2 \) (from \( P(x) \)).
- For \( (x+2) \), it appears only in \( P(x) \).
- For \( (x-2) \), it appears only in \( Q(x) \).

Thus, the LCM is:
\[ \text{LCM} = (x+1)^2 (x+2)(x-2) \]

### Final Results
- **HCF**: \( (x+1) \)
- **LCM**: \( (x+1)^2 (x+2)(x-2) \)

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