Let's perform the indicated operation and simplify the result step-by-step. We'll start by simplifying the numerator and the denominator separately and then combine them.
Step 1: Simplify the Numerator
The numerator is given by:
[tex]\[
\frac{x-8}{x+8} + \frac{x-1}{x+1}
\][/tex]
To add these fractions, we need a common denominator. The common denominator for [tex]\( x+8 \)[/tex] and [tex]\( x+1 \)[/tex] is [tex]\((x+8)(x+1)\)[/tex]:
[tex]\[
\frac{x-8}{x+8} + \frac{x-1}{x+1} = \frac{(x-8)(x+1) + (x-1)(x+8)}{(x+8)(x+1)}
\][/tex]
Now, expand and simplify the numerators:
[tex]\[
(x-8)(x+1) = x^2 + x - 8x - 8 = x^2 - 7x - 8
\][/tex]
[tex]\[
(x-1)(x+8) = x^2 + 8x - x - 8 = x^2 + 7x - 8
\][/tex]
Add these results:
[tex]\[
\frac{x^2 - 7x - 8 + x^2 + 7x - 8}{(x+8)(x+1)} = \frac{2x^2 - 16}{(x+8)(x+1)}
\][/tex]
Factor the numerator:
[tex]\[
2x^2 - 16 = 2(x^2 - 8) = 2(x - \sqrt{8})(x + \sqrt{8})
\][/tex]
So the numerator in factored form is:
[tex]\[
\frac{2(x^2 - 8)}{(x+8)(x+1)}
\][/tex]
Step 2: Simplify the Denominator
The denominator is given by:
[tex]\[
\frac{x}{x+1} - \frac{5x-1}{x}
\][/tex]
Find a common denominator for [tex]\( x+1 \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
\frac{x}{x+1} - \frac{5x-1}{x} = \frac{x \cdot x - (5x-1)(x+1)}{x(x+1)}
\][/tex]
Expand and simplify the numerator:
[tex]\[
x \cdot x = x^2
\][/tex]
[tex]\[
(5x-1)(x+1) = 5x^2 + 5x - x - 1 = 5x^2 + 4x - 1
\][/tex]
Subtract these results:
[tex]\[
\frac{x^2 - (5x^2 + 4x - 1)}{x(x+1)} = \frac{x^2 - 5x^2 - 4x + 1}{x(x+1)} = \frac{-4x^2 - 4x + 1}{x(x+1)}
\][/tex]
So the denominator in factored form is:
[tex]\[
\frac{-4x^2 - 4x + 1}{x(x+1)}
\][/tex]
Step 3: Simplify the Entire Expression
Now we combine the simplified numerator and denominator:
[tex]\[
\frac{\frac{2(x^2 - 8)}{(x+8)(x+1)}}{\frac{-4x^2 - 4x + 1}{x(x+1)}} = \frac{2(x^2 - 8) \cdot x(x+1)}{(x+8)(x+1) \cdot (-4x^2 - 4x + 1)}
\][/tex]
Cancel the common factors [tex]\((x+1)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{2(x^2 - 8) \cdot x}{(x+8) \cdot (-4x^2 - 4x + 1)}
\][/tex]
Therefore, the simplified result of the given expression, in its factored form, is:
[tex]\[
\boxed{\frac{2x(x^2 - 8)}{(x+8)(-4x^2 - 4x + 1)}}
\][/tex]
