What is the least common denominator of the rational expressions x-1/x^2+8x+7 and -3/x^2+x

To find the least common denominator (LCD) of two rational expressions, we need to factor the denominators of each expression and find the product of all the unique factors.

Let's start by factoring the denominators:

For the expression x-1/x^2+8x+7:
The denominator x^2+8x+7 is a quadratic expression. We can factor it as (x+7)(x+1).

For the expression -3/x^2+x:
The denominator x^2+x is also a quadratic expression. We can factor it as x(x+1).

Now, let's identify the unique factors for the LCD:
1. (x+7) from the first expression.
2. (x+1) from the first expression.
3. x from the second expression.

To find the LCD, we take the product of all the unique factors:
LCD = (x+7)(x+1)x

Therefore, the least common denominator of the rational expressions x-1/x^2+8x+7 and -3/x^2+x is (x+7)(x+1)x.