What is the value of `((x ^(2) -x - 6)/(x ^(2) + x - 12)) ÷ ((x ^(2) + 5x + 6)/(x ^(2) + 7x + 12))` ?

Options

A. 1
B. `((x- 3))/((x+ 3))`
C. `((x+ 4))/(( x - 3))`
D. `((x - 3))/( (x + 4))`

Answer

To solve the expression

\[ \frac{(x^2 - x - 6)}{(x^2 + x - 12)} \div \frac{(x^2 + 5x + 6)}{(x^2 + 7x + 12)}, \]

we can follow these steps:

### Step 1: Rewrite the Division as Multiplication
We can rewrite the division of fractions as multiplication by the reciprocal:

\[ \frac{(x^2 - x - 6)}{(x^2 + x - 12)} \times \frac{(x^2 + 7x + 12)}{(x^2 + 5x + 6)}. \]

### Step 2: Factor Each Quadratic Expression
Now, we will factor each of the quadratic expressions:

1. **Factor \(x^2 - x - 6\)**:
- We look for two numbers that multiply to \(-6\) and add to \(-1\). The numbers are \(-3\) and \(2\).
- Thus, \(x^2 - x - 6 = (x - 3)(x + 2)\).

2. **Factor \(x^2 + x - 12\)**:
- We look for two numbers that multiply to \(-12\) and add to \(1\). The numbers are \(4\) and \(-3\).
- Thus, \(x^2 + x - 12 = (x + 4)(x - 3)\).

3. **Factor \(x^2 + 5x + 6\)**:
- We look for two numbers that multiply to \(6\) and add to \(5\). The numbers are \(2\) and \(3\).
- Thus, \(x^2 + 5x + 6 = (x + 2)(x + 3)\).

4. **Factor \(x^2 + 7x + 12\)**:
- We look for two numbers that multiply to \(12\) and add to \(7\). The numbers are \(3\) and \(4\).
- Thus, \(x^2 + 7x + 12 = (x + 3)(x + 4)\).

### Step 3: Substitute the Factored Forms
Now we substitute the factored forms back into the expression:

\[ \frac{(x - 3)(x + 2)}{(x + 4)(x - 3)} \times \frac{(x + 3)(x + 4)}{(x + 2)(x + 3)}. \]

### Step 4: Cancel Common Factors
Now we can cancel out the common factors in the numerator and denominator:

- \(x - 3\) cancels with \(x - 3\).
- \(x + 2\) cancels with \(x + 2\).
- \(x + 3\) cancels with \(x + 3\).
- \(x + 4\) cancels with \(x + 4\).

After canceling, we are left with:

\[ 1. \]

### Final Answer
Thus, the value of the original expression is:

\[ \boxed{1}. \]