ANSWER
[tex] \frac{f + g}{f - g} = \frac{2 {x}^{2} - 85 }{ 13} [/tex]
EXPLANATION
The given functions are
[tex]f(x) = \frac{x - 6}{x + 7} [/tex]
and
[tex]g(x) = \frac{x - 7}{x + 6}[/tex]
We are required to simplify,
[tex] \frac{f + g}{f - g} [/tex]
We proceed as follows:
[tex] \frac{f + g}{f - g} = \frac{f (x)+ g(x)}{f(x) - g(x)} [/tex]
[tex] \frac{f + g}{f - g} = \frac{ \frac{x - 6}{x + 7} + \frac{x - 7}{x + 6}}{\frac{x - 6}{x + 7} - \frac{x - 7}{x + 6}}[/tex]
This gives us,
[tex] \frac{f + g}{f - g} = \frac{ \frac{(x - 6)(x + 6) + (x + 7)(x - 7)}{(x + 7)(x + 6)} }{\frac{(x - 6)(x + 6) - (x + 7)(x - 7)}{(x + 7)(x + 6)} }[/tex]
This simplifies to,
[tex] \frac{f + g}{f - g} = \frac{(x + 6)(x - 6) + (x + 7)(x - 7}{(x - 6)(x + 6) - (x - 7)x + 7)} [/tex]
We now apply difference of two squares to get,
[tex] \frac{f + g}{f - g} = \frac{ {x}^{2} - 36 + {x}^{2} - 49}{ {x}^{2} - 36- ( {x}^{2} - 49)} [/tex]
[tex] \frac{f + g}{f - g} = \frac{ {x}^{2} - 36 + {x}^{2} - 49}{ {x}^{2} - 36- {x}^{2} + 49} [/tex]
This further simplifies to,
[tex] \frac{f + g}{f - g} = \frac{2 {x}^{2} - 85 }{ 13} [/tex]
Therefore the correct answer is C
