Solve `(x^2+3x+2)/(x^2-6x-7)=0.`

Explanation

To solve the equation \(\frac{x^2 + 3x + 2}{x^2 - 6x - 7} = 0\), we follow these steps:

### Step 1: Understand when the fraction equals zero
A fraction equals zero when its numerator is zero (as long as the denominator is not zero). Therefore, we need to set the numerator equal to zero:

\[ x^2 + 3x + 2 = 0 \]

### Step 2: Factor the numerator
Next, we factor the quadratic equation \(x^2 + 3x + 2\):

\[ (x + 1)(x + 2) = 0 \]

### Step 3: Solve for \(x\)
Setting each factor equal to zero gives us:

1. \(x + 1 = 0 \implies x = -1\)
2. \(x + 2 = 0 \implies x = -2\)

Thus, the potential solutions are \(x = -1\) and \(x = -2\).

### Step 4: Check the denominator
Next, we must ensure that these solutions do not make the denominator zero. We set the denominator equal to zero:

\[ x^2 - 6x - 7 = 0 \]

### Step 5: Factor the denominator
We can factor this quadratic as follows:

\[ (x - 7)(x + 1) = 0 \]

### Step 6: Solve for \(x\)
Setting each factor equal to zero gives us:

1. \(x - 7 = 0 \implies x = 7\)
2. \(x + 1 = 0 \implies x = -1\)

### Step 7: Identify restrictions
The values \(x = 7\) and \(x = -1\) make the denominator zero. Therefore, these values are not allowed in the solution set.

### Step 8: Determine the valid solution
From our earlier solutions \(x = -1\) and \(x = -2\), we see that \(x = -1\) is not valid since it makes the denominator zero. Thus, the only acceptable solution is:

\[ x = -2 \]

### Final Solution
The solution to the equation \(\frac{x^2 + 3x + 2}{x^2 - 6x - 7} = 0\) is:

\[ \boxed{-2} \]