Question

Add the rational expressions to find the sum: −4/x−4+x^2−2x/2x−8
. Put the answer into its simplest form.(1 point)
Responses

x2−2x−82x−8
Start Fraction x squared minus 2 x minus 8 over 2 x minus 8 End Fraction

x2−2x2x
Start Fraction x squared minus 2 x over 2 x End Fraction

x+22
Start Fraction x plus 2 over 2 End Fraction

x2−2x−4x−4

Answer 1

To find the sum of the rational expressions \(-\frac{4}{x} - \frac{4+x^2-2x}{2x-8}\), we first simplify the expression.

1. **Factor the denominator**:
The denominator \(2x - 8\) can be factored as \(2(x - 4)\).

2. **Rewrite the expression**:
The original expression becomes:
\[
-\frac{4}{x} - \frac{4 + x^2 - 2x}{2(x - 4)}
\]
Now simplify the second term:
\[
-\frac{4 + x^2 - 2x}{2(x - 4)} = -\frac{x^2 - 2x + 4}{2(x - 4)}
\]

3. **Finding a common denominator**:
The common denominator for \(x\) and \(2(x - 4)\) is \(2x(x - 4)\).

4. **Rewrite each term with the common denominator**:
\[
-\frac{4}{x} = -\frac{4 \cdot 2(x-4)}{2x(x-4)} = -\frac{8(x-4)}{2x(x-4)}
\]
For the second term, it’s already over the common denominator:
\[
-\frac{x^2 - 2x + 4}{2(x - 4)} = -\frac{x^2 - 2x + 4}{2(x - 4)} = -\frac{x^2 - 2x + 4+x^2-2x + 8}{2x(x - 4)}
\]

5. **Combine the fractions**:
Combine the two fractions:
\[
\frac{-8(x-4) - (x^2 - 2x + 4)}{2x(x-4)}
\]

6. **Simplify the numerator**:
\[
-8(x-4) = -8x + 32
\]
So:
\[-8x + 32 - (x^2 - 2x + 4)\]
\[
= -8x + 32 - x^2 + 2x - 4 = -x^2 - 6x + 28
\]

7. **Final expression**:
Thus, the result is:
\[
\frac{-x^2 - 6x + 28}{2x(x-4)}
\]

8. **Simplify further if possible**:
The expression \(-x^2 - 6x + 28\) can be rearranged as:
\[
-(x^2 + 6x - 28) = -(x + 14)(x - 2) \text{ (factoring)}
\]

So the sum of the rational expressions simplified is:
\[
\frac{-(x + 14)(x - 2)}{2x(x - 4)}
\]

### The answer in simplified form:
The expression isn't present among the provided options. However, reducing gives you the essential information, and the closest simplified expression would be:
\[
\frac{-(x^2 + 6x - 28)}{2x(x-4)}
\]

### Please match to your options:
1. \(\frac{x^2 - 2x - 8}{2x - 8}\)
2. \(\frac{x^2 - 2x}{2x}\)
3. \(\frac{x + 2}{2}\)
4. \(-4x - 4\)

From the detail explored, no exact matches are found among these options, possibly needing a re-evaluation based on the initial presentation.