e. \( P(Q+6)=P-12 \quad \) for \( P \)

To solve the equation \( P(Q+6)=P-12 \), we can start by expanding the left side: \( PQ + 6P = P - 12 \). Next, we can rearrange the equation to get all terms involving \( P \) on one side, leading to \( PQ + 6P - P = -12 \). Simplifying gives us \( PQ + 5P = -12 \). Now, if we take \( P \) as a common factor, we have \( P(Q + 5) = -12 \). Finally, we can isolate \( P \) by dividing both sides by \( Q + 5 \), yielding \( P = \frac{-12}{Q + 5} \). Voila! You have expressed \( P \) in terms of \( Q \).