If the LCM and HCF of two expressions are `(x^(2)+6x+8)(x+1)` and `(x+1)` respectively & one of the expression being `x^(2)+3x+2` then find the another expression ?

Options

A. `x^(2)+5x+4`
B. `x^(2)-5x+4`
C. `x^(2)+4x+5`
D. `x^(2)-4x+5`

Answer

To find the second expression given the LCM and HCF, we will follow these steps:

### Step 1: Factor the given expressions
We start by factoring the LCM and the known expression.

1. **Factor the LCM**:
\[ \text{LCM} = (x^2 + 6x + 8)(x + 1) \]
First, we factor \(x^2 + 6x + 8\):
\[ x^2 + 6x + 8 = (x + 4)(x + 2) \]
So, the LCM can be rewritten as:
\[ \text{LCM} = (x + 4)(x + 2)(x + 1) \]

2. **Factor the known expression**:
\[ x^2 + 3x + 2 = (x + 2)(x + 1) \]

### Step 2: Use the relationship between LCM, HCF, and the expressions
We know that:
\[ \text{LCM} \times \text{HCF} = \text{Expression 1} \times \text{Expression 2} \]
Given:
- LCM = \((x + 4)(x + 2)(x + 1)\)
- HCF = \((x + 1)\)
- Expression 1 = \(x^2 + 3x + 2 = (x + 2)(x + 1)\)

Let the second expression be \(E_2\). Thus, we can write:
\[ \text{LCM} \times \text{HCF} = (x + 4)(x + 2)(x + 1) \times (x + 1) \]
This simplifies to:
\[ (x + 4)(x + 2)(x + 1)^2 \]

### Step 3: Set up the equation
Now, we have:
\[ (x + 2)(x + 1) \times E_2 = (x + 4)(x + 2)(x + 1)^2 \]

### Step 4: Solve for the second expression \(E_2\)
To find \(E_2\), we can divide both sides by \((x + 2)(x + 1)\):
\[ E_2 = \frac{(x + 4)(x + 2)(x + 1)^2}{(x + 2)(x + 1)} \]
This simplifies to:
\[ E_2 = (x + 4)(x + 1) \]

### Step 5: Expand \(E_2\)
Now, we expand \(E_2\):
\[ E_2 = (x + 4)(x + 1) = x^2 + 5x + 4 \]

### Conclusion
Thus, the second expression is:
\[ \boxed{x^2 + 5x + 4} \]