Question

Simplify the complex fraction.
StartStartFraction StartFraction 2 Over p plus 2 EndFraction minus 1 OverOver StartFraction 2 Over p plus 2 EndFraction plus 1 EndEndFraction

Answer 1

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

\[
\frac{-p}{p + 4}.
\]