Question

Which answer is correct? What is the LCD in its factored from of the following rational expression subtraction?
x2+2x/x2+7x-8 - x/x-1

(x-1)(x+8)
x(x+2)
(x-1)2(x+8
(x-1)(x2+7x-8)

Answer 1

To find the least common denominator (LCD) for the given rational expressions, we need to factor the denominators completely.

The two rational expressions are:
1. \(\frac{x^2 + 2x}{x^2 + 7x - 8}\)
2. \(\frac{x}{x-1}\)

### Step 1: Factor the denominators

1. **Denominator of the first expression**: \(x^2 + 7x - 8\)

We want to factor \(x^2 + 7x - 8\). We need to find two numbers that multiply to \(-8\) and add to \(7\). The factorization is:
\[
x^2 + 7x - 8 = (x + 8)(x - 1)
\]
So, the first denominator factors to \((x + 8)(x - 1)\).

2. **Denominator of the second expression**: \(x - 1\)

This is already factored.

### Step 2: Determine the Least Common Denominator (LCD)

The least common denominator of both expressions must include all unique factors from both denominators.

- From \((x + 8)(x - 1)\), we have:
- \(x + 8\)
- \(x - 1\)

- From \(x - 1\), we also have just \(x - 1\) (which is already included).

Therefore, the LCD is:
\[
(x - 1)(x + 8)
\]

### Conclusion

The correct answer for the least common denominator (LCD) in its factored form is:
\((x - 1)(x + 8)\)

Thus, the correct option among the given choices is:
**(x-1)(x+8)**