Is k^2 + k - 2 > 0 ?

(1) k is greater than zero
(2) k is an even integer

why exactly is it justified to go from asking whether the inequality equation is greater than zero to setting k = to 0? and assuming the factors are
relevant? It seems to me that it\'s a bit of a stretch.

The question asks if the equation is greater than zero, not if its equal to zero. There is a distinction.
For example, if we factor the left hand side of the equation we have (k+2)(k-1) = 0 so if k is equal to 1 or , -2 the equation is equal to zero. However, in order for the equation to be greater than zero, k cannot be 1 or -2. Those are not the solutions to (k+2)(k-1) > 0.

It can be usefull to draw a number line for better comprehension:

Please, remember that k² + k – 2 = 0 when k = -2 or k = 1.