Final answer:
The trinomial can be factored as (2m - 3)(m + 1). The integers that make this trinomial factorable are -3 and 2.
Step-by-step explanation:
This trinomial can be factored by looking for two numbers whose product is -6 and whose sum is the coefficient of the middle term, which is k. In this case, the numbers are -3 and 2. So, we can factor the trinomial as follows:
2m² + km - 3 = (2m - 3)(m + 1)
Therefore, all integers k that can make the trinomial factorable are -3 and 2.