To simplify the expression \(\frac{f + g}{f - g}\) where
\[ f(x) = \frac{x - 6}{x + 7} \quad \text{and} \quad g(x) = \frac{x - 7}{x + 6}, \]
we first need to calculate \(f + g\) and \(f - g\).
Step 1: Find \(f + g\)
\[ f + g = \frac{x - 6}{x + 7} + \frac{x - 7}{x + 6}. \]
To add these fractions, we need a common denominator, which is \((x + 7)(x + 6)\):
\[ f + g = \frac{(x - 6)(x + 6) + (x - 7)(x + 7)}{(x + 7)(x + 6)}. \]
Calculating the numerators:
- For \((x - 6)(x + 6)\):
\[ (x - 6)(x + 6) = x^2 - 36. \]
- For \((x - 7)(x + 7)\):
\[ (x - 7)(x + 7) = x^2 - 49. \]
Now combine these results:
\[ f + g = \frac{x^2 - 36 + x^2 - 49}{(x + 7)(x + 6)} = \frac{2x^2 - 85}{(x + 7)(x + 6)}. \]
Step 2: Find \(f - g\)
\[ f - g = \frac{x - 6}{x + 7} - \frac{x - 7}{x + 6}. \]
Again, using a common denominator:
\[ f - g = \frac{(x - 6)(x + 6) - (x - 7)(x + 7)}{(x + 7)(x + 6)}. \]
We already calculated the numerators earlier:
\[ f - g = \frac{(x^2 - 36) - (x^2 - 49)}{(x + 7)(x + 6)} = \frac{-36 + 49}{(x + 7)(x + 6)} = \frac{13}{(x + 7)(x + 6)}. \]
Step 3: Combine \(f + g\) and \(f - g\) to find \(\frac{f + g}{f - g}\)
Now we can calculate:
\[ \frac{f + g}{f - g} = \frac{\frac{2x^2 - 85}{(x + 7)(x + 6)}}{\frac{13}{(x + 7)(x + 6)}}. \]
This simplifies to:
\[ \frac{2x^2 - 85}{13}. \]
Thus, the final simplified expression is:
\[ \frac{2x^2 - 85}{13}. \]
So the correct answer is:
\[ \boxed{\frac{2x^2 - 85}{13}}. \]