IF `(x^2+x-6)/(x^2-8x+12)=4`, What is the value of x?

Answer

To solve the equation

\[ \frac{x^2 + x - 6}{x^2 - 8x + 12} = 4, \]

we will follow these steps:

### Step 1: Eliminate the denominator
Multiply both sides of the equation by the denominator \(x^2 - 8x + 12\):

\[ x^2 + x - 6 = 4(x^2 - 8x + 12). \]

### Step 2: Expand the right side
Now, expand the right side:

\[ x^2 + x - 6 = 4x^2 - 32x + 48. \]

### Step 3: Rearrange the equation
Next, move all terms to one side of the equation to set it to zero:

\[ x^2 + x - 6 - 4x^2 + 32x - 48 = 0. \]

Combine like terms:

\[ -3x^2 + 33x - 54 = 0. \]

### Step 4: Simplify the equation
To make the equation simpler, divide all terms by -3:

\[ x^2 - 11x + 18 = 0. \]

### Step 5: Factor the quadratic equation
Now, we will factor the quadratic equation:

\[ x^2 - 11x + 18 = (x - 2)(x - 9) = 0. \]

### Step 6: Solve for x
Set each factor equal to zero:

1. \(x - 2 = 0 \Rightarrow x = 2\)
2. \(x - 9 = 0 \Rightarrow x = 9\)

### Step 7: Check for extraneous solutions
We need to check if either solution makes the denominator zero. The denominator is:

\[ x^2 - 8x + 12. \]

Substituting \(x = 2\):

\[ 2^2 - 8(2) + 12 = 4 - 16 + 12 = 0. \]

Since this makes the denominator zero, \(x = 2\) is not a valid solution.

Now substituting \(x = 9\):

\[ 9^2 - 8(9) + 12 = 81 - 72 + 12 = 21. \]

This does not make the denominator zero, so \(x = 9\) is a valid solution.

### Final Answer
The value of \(x\) is:

\[ \boxed{9}. \]