Express the following in the lowest terms.
`sqrt(((x^(2)+3x+2)(x^(2)+5x+6))/(x^(2)(x^(2)+4x+3)))`

Options

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

Answer

To express the given expression in the lowest terms, we will follow these steps:

### Step 1: Factor the Numerator and Denominator
We start with the expression under the square root:
\[ \sqrt{\frac{(x^2 + 3x + 2)(x^2 + 5x + 6)}{x^2(x^2 + 4x + 3)}} \]

First, we factor each polynomial.

1. **Factor \(x^2 + 3x + 2\)**:
- This can be factored as \((x + 1)(x + 2)\).

2. **Factor \(x^2 + 5x + 6\)**:
- This can be factored as \((x + 2)(x + 3)\).

3. **Factor \(x^2 + 4x + 3\)**:
- This can be factored as \((x + 1)(x + 3)\).

Now, substituting these factorizations into the expression gives us:
\[ \sqrt{\frac{((x + 1)(x + 2))((x + 2)(x + 3))}{x^2((x + 1)(x + 3))}} \]

### Step 2: Simplify the Expression
Now we can simplify the expression:
\[ \sqrt{\frac{(x + 1)(x + 2)^2(x + 3)}{x^2(x + 1)(x + 3)}} \]

### Step 3: Cancel Common Factors
We can cancel out the common factors in the numerator and denominator:
- Cancel \((x + 1)\) from both numerator and denominator.
- Cancel \((x + 3)\) from both numerator and denominator.

This simplifies to:
\[ \sqrt{\frac{(x + 2)^2}{x^2}} \]

### Step 4: Simplify 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 Result
Thus, the expression in the lowest terms is:
\[ \frac{x + 2}{x} \]