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 simplify the expression step by step.

The expression we need to simplify is:

\[ \sqrt{\frac{(x^2 + 3x + 2)(x^2 + 5x + 6)}{x^2(x^2 + 4x + 3)}} \]

### Step 1: Factor the Numerator and Denominator

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 \((x + 1)(x + 3)\).

So, we can rewrite the expression as:

\[ \sqrt{\frac{((x + 1)(x + 2))((x + 2)(x + 3))}{x^2((x + 1)(x + 3))}} \]

### Step 2: Simplify the Expression

Now, substituting the factored forms into the expression gives us:

\[ \sqrt{\frac{(x + 1)(x + 2)(x + 2)(x + 3)}{x^2(x + 1)(x + 3)}} \]

Next, we can cancel out the common factors in the numerator and the denominator:

- The \((x + 1)\) in the numerator and denominator cancels out.
- The \((x + 3)\) in the numerator and denominator cancels out.

This simplifies our expression to:

\[ \sqrt{\frac{(x + 2)^2}{x^2}} \]

### Step 3: 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:

\[ \frac{x + 2}{x} \]