For 8.23, example 8.3 may give you a hint on where to start.

For 8.26, try rewriting the equation in the form $(x-a)(y-b) = c$. Then factor $c$ as the product of two integers; these factors are precisely $(x-a)$ and $(y-b)$. So you can find one solution $(x,y)$ for each way to factor $c$ as the product of two integers.

For 8.27, example 8.6 shows there is exactly one solution over the positive integers. Try to use this solution to find a second solution where the numbers may be negative.