What should be subrated from `(7x)/(x^(2) + x - 12)` to get `(8)/(x + 4)` ?

Options

A. `(2)/((x - 3))`
B. `(2)/((2x - 3))`
C. `(x + 1)/((x - 3))`
D. `(24-x)/(x^(2) + x - 12)`

Answer

To solve the problem of what should be subtracted from \(\frac{7x}{x^2 + x - 12}\) to get \(\frac{8}{x + 4}\), we can follow these steps:

### Step 1: Set up the equation
We need to find a value \(A\) such that:
\[ \frac{7x}{x^2 + x - 12} - A = \frac{8}{x + 4} \]
Rearranging this gives us:
\[ A = \frac{7x}{x^2 + x - 12} - \frac{8}{x + 4} \]

### Step 2: Factor the denominator
The denominator \(x^2 + x - 12\) can be factored. We look for two numbers that multiply to \(-12\) and add to \(1\). The numbers \(4\) and \(-3\) work, so:
\[ x^2 + x - 12 = (x + 4)(x - 3) \]

### Step 3: Rewrite the equation
Now, we can rewrite \(A\) using the factored form:
\[ A = \frac{7x}{(x + 4)(x - 3)} - \frac{8}{x + 4} \]

### Step 4: Find a common denominator
The common denominator for the two fractions is \((x + 4)(x - 3)\). We can rewrite the second fraction:
\[ \frac{8}{x + 4} = \frac{8(x - 3)}{(x + 4)(x - 3)} \]

### Step 5: Combine the fractions
Now we can combine the fractions:
\[ A = \frac{7x - 8(x - 3)}{(x + 4)(x - 3)} \]
Expanding the numerator:
\[ A = \frac{7x - 8x + 24}{(x + 4)(x - 3)} = \frac{-x + 24}{(x + 4)(x - 3)} \]

### Step 6: Simplify the expression
Thus, the expression for \(A\) simplifies to:
\[ A = \frac{24 - x}{(x + 4)(x - 3)} \]

### Final Answer
The value that should be subtracted from \(\frac{7x}{x^2 + x - 12}\) to get \(\frac{8}{x + 4}\) is:
\[ A = \frac{24 - x}{(x + 4)(x - 3)} \]

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