The Zero Product Property is a property of real numbers that says that for any two real numbers x and y, if x*y = 0 then x = 0 or y = 0 (or both).
Thus if we have (z-2)(z+1) = 0, then we know that either z - 2 = 0, or z + 1 = 0, or both.
If z - 2 = 0, z = 2. That's one solution.
If z + 1 = 0, z = -1. That's another solution.
Both can't be true at the same time, so that's not a solution.
This is why we factor polynomials to find roots. If you can break up an expression into factors and the product of all the factors is equal to zero, you know that at least one factor it itself zero.