To simplify the complex fraction
\[ \frac{\frac{2}{p+2} - 1}{\frac{2}{p+2} + 1}, \]
we will first handle the numerator and denominator separately.
Step 1: Simplify the numerator
The numerator is
\[ \frac{2}{p+2} - 1. \]
To combine these terms, we can express \(1\) as \(\frac{p+2}{p+2}\):
\[ \frac{2}{p+2} - 1 = \frac{2}{p+2} - \frac{p+2}{p+2} = \frac{2 - (p + 2)}{p + 2} = \frac{2 - p - 2}{p + 2} = \frac{-p}{p + 2}. \]
Step 2: Simplify the denominator
The denominator is
\[ \frac{2}{p+2} + 1. \]
Similarly, express \(1\) as \(\frac{p+2}{p+2}\):
\[ \frac{2}{p+2} + 1 = \frac{2}{p+2} + \frac{p+2}{p+2} = \frac{2 + (p + 2)}{p + 2} = \frac{2 + p + 2}{p + 2} = \frac{p + 4}{p + 2}. \]
Step 3: Combine the simplified numerator and denominator
Now we substitute our results back into the complex fraction:
\[ \frac{\frac{-p}{p + 2}}{\frac{p + 4}{p + 2}}. \]
Step 4: Simplify the fraction
When dividing by a fraction, we multiply by its reciprocal:
\[ \frac{-p}{p + 2} \cdot \frac{p + 2}{p + 4} = \frac{-p (p + 2)}{(p + 2)(p + 4)}. \]
Since \(p + 2\) cancels out (given \(p \neq -2\)):
\[ \frac{-p}{p + 4}. \]
Final Result
Thus, the simplified form of the complex fraction is
\[ \frac{-p}{p + 4}. \]