f. \( \frac{1}{p}+\frac{2}{p+4}=\frac{5}{p} \)

To solve the equation \( \frac{1}{p} + \frac{2}{p+4} = \frac{5}{p} \), first, let's simplify it. We can subtract \( \frac{1}{p} \) from both sides: \[ \frac{2}{p+4} = \frac{5}{p} - \frac{1}{p} \] This simplifies to: \[ \frac{2}{p+4} = \frac{4}{p} \] Next, we cross-multiply: \[ 2p = 4(p + 4) \] Expanding the right side: \[ 2p = 4p + 16 \] Now, we can isolate \( p \) by subtracting \( 4p \) from both sides: \[ 2p - 4p = 16 \] This gives us: \[ -2p = 16 \] Dividing by -2, we find: \[ p = -8 \] To verify, substitute \( p = -8 \) back into the original equation: \[ \frac{1}{-8} + \frac{2}{-4} = \frac{5}{-8} \] Calculating each term gives: \[ -\frac{1}{8} - \frac{1}{2} = -\frac{1}{8} - \frac{4}{8} = -\frac{5}{8} \] The right side is: \[ -\frac{5}{8} \] Both sides match, confirming the solution. Thus, \( p = -8 \) is the final answer!