Step 1: Use the coefficients-sum-to-zero rule as the starting point. Any Laplacian-type sharpening mask is a high-pass filter, and its coefficients must sum to 0 so a flat, constant-brightness region produces zero output. Step 2: Build the 4-neighbour version. The simplest Laplacian mask places $+1$ at the 4 pixels directly above, below, left and right of the center, and $0$ at the 4 diagonal corners. For the total to be zero: $center+(1+1+1+1)=0 \Rightarrow center=-4$. This gives option (A) as valid. Step 3: Build the 8-neighbour version. A stronger mask places $+1$ at all 8 surrounding pixels. For the total to be zero: $center+(1\times8)=0 \Rightarrow center=-8$. This gives option (B) as valid. Step 4: Show -2 and -6 cannot satisfy the zero-sum rule with either standard mask. With unit neighbour weights, only 4 or 8 neighbouring unit coefficients are used in the two textbook masks, forcing the center to be exactly $-4$ or $-8$; there is no standard unit-weighted Laplacian construction balancing to $-2$ or $-6$. Step 5: Final answer. The possible center-pixel values are $-4$ and $-8$.\[ \boxed{-4 \text{ and } -8} \]
