`sqrt(((x^2 + 3x + 2)(x^2 + 5x + 6))/(x^2 (x^2 + 4x+ 3)))` is equal to :

Options

A. `x(x+1)`
B. `(x + 2)/x`
C. `x/( x + 2)`
D. `x (x + 2)`

Answer

To solve the expression \( \sqrt{\frac{(x^2 + 3x + 2)(x^2 + 5x + 6)}{x^2 (x^2 + 4x + 3)}} \), we will follow these steps:

### Step 1: Factor the Quadratic Expressions
First, we need to factor the quadratic expressions in the numerator and the denominator.

1. **Factor \( x^2 + 3x + 2 \)**:
- We look for two numbers that multiply to \( 2 \) (the constant term) and add up to \( 3 \) (the coefficient of \( x \)).
- The numbers \( 1 \) and \( 2 \) work:
\[ x^2 + 3x + 2 = (x + 1)(x + 2) \]

2. **Factor \( x^2 + 5x + 6 \)**:
- We look for two numbers that multiply to \( 6 \) and add up to \( 5 \).
- The numbers \( 2 \) and \( 3 \) work:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]

3. **Factor \( x^2 + 4x + 3 \)**:
- We look for two numbers that multiply to \( 3 \) and add up to \( 4 \).
- The numbers \( 1 \) and \( 3 \) work:
\[ x^2 + 4x + 3 = (x + 1)(x + 3) \]

### Step 2: Substitute the Factors into the Expression
Now we can substitute the factored forms back into the original expression:
\[ \sqrt{\frac{(x + 1)(x + 2)(x + 2)(x + 3)}{x^2 (x + 1)(x + 3)}} \]

### Step 3: Simplify the Expression
Next, we can simplify the expression by canceling out common factors in the numerator and denominator:
- The factors \( (x + 1) \) and \( (x + 3) \) cancel out:
\[ \sqrt{\frac{(x + 2)^2}{x^2}} \]

### Step 4: Apply the Square Root
Now, we can take the square root of the simplified expression:
\[ \sqrt{\frac{(x + 2)^2}{x^2}} = \frac{x + 2}{x} \]

### Final Answer
Thus, the expression simplifies to:
\[ \frac{x + 2}{x} \]

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