What are three consecutive even integers whose sum is 32?

That isn't possible; three consecutive integers, or three consecutive positive integers, always have a sum that is a multiple of 3.

In general, you can solve this quickly by trial and error. In this case, you will quickly find that a certain set of three consecutive integers will give you a sum that is TOO LOW, while the next-higher even integers will give you a sum that is TOO HIGH.

You can also write an equation and solve it: n + (n + 2) + (n + 4) = 32. If you solve it, you will find that the solution is fractional, not integral.