Simplify the following expression: $\frac{\frac{3}{p^2} - \frac{1}{p^2-4}}{1 - \frac{6}{p^2}}$

1. Problem Description

Simplify the following expression:

$\frac{\frac{3}{p^2} - \frac{1}{p^2-4}}{1 - \frac{6}{p^2}}$

2. Solution Steps

First, we simplify the numerator:

$\frac{3}{p^2} - \frac{1}{p^2-4} = \frac{3(p^2-4) - 1(p^2)}{p^2(p^2-4)} = \frac{3p^2 - 12 - p^2}{p^2(p^2-4)} = \frac{2p^2 - 12}{p^2(p^2-4)} = \frac{2(p^2-6)}{p^2(p^2-4)}$

Next, we simplify the denominator:

$1 - \frac{6}{p^2} = \frac{p^2}{p^2} - \frac{6}{p^2} = \frac{p^2 - 6}{p^2}$

Now we divide the simplified numerator by the simplified denominator:

$\frac{\frac{2(p^2-6)}{p^2(p^2-4)}}{\frac{p^2-6}{p^2}} = \frac{2(p^2-6)}{p^2(p^2-4)} \cdot \frac{p^2}{p^2-6} = \frac{2(p^2-6)p^2}{p^2(p^2-4)(p^2-6)}$

We can cancel out the common factors $p^2$ and $(p^2-6)$, assuming $p^2 \ne 0$ and $p^2 \ne 6$.

$\frac{2}{p^2-4} = \frac{2}{(p-2)(p+2)}$

3. Final Answer

$\frac{2}{p^2-4}$