Explanation:

To determine the correct option for incrementing the Edit_Count__c field on a Case object whenever it is changed, we need to analyze the provided Apex trigger and class implementations. The trigger is a before update trigger on the Case object, and the goal is to ensure the Edit_Count__c field increments only once per update operation, avoiding multiple increments due to recursive trigger calls. Let’s evaluate each option step-by-step.

Trigger and Requirement Analysis

The trigger is defined as:
trigger on Case (before update) {
CaseTriggerHandler.handle(Trigger.new);
}

This trigger calls a static method handle in the CaseTriggerHandler class, passing the list of new Case records (Trigger.new). The handle method is responsible for incrementing the Edit_Count__c field. Since it’s a before update trigger, the field can be modified directly on the records in Trigger.new without requiring a separate DML operation. However, a common challenge with triggers is recursion—e.g., updating the record within the trigger could cause it to fire again, leading to unintended multiple increments. We need a mechanism to prevent this, ensuring the counter increments only once per update.

Evaluating the Options

Option A:
Analysis: This implementation simply loops through the cases and increments Edit_Count__c by 1 for each record. However, it lacks any recursion control. If the trigger update causes another update (e.g., via a workflow or process), the trigger will re-execute, incrementing the counter multiple times. This does not meet the requirement of a single increment per change.
Verdict: Incorrect due to potential recursion issues.

Option B:
Analysis: This approach uses a firstRun Boolean to control execution. The trigger sets firstRun = true before calling handle, and the handle method checks this flag to perform the increment only if firstRun is true, then sets it to false. However, there are issues:

➜ firstRun is an instance variable in the class, but the handle method is static. Static methods cannot directly access instance variables unless they are also static or passed as parameters, which isn’t done here. This code would not compile due to this mismatch.
➜ Even if made static (e.g., static Boolean firstRun), the trigger and class logic would need to align properly. The current setup has redundant checks and could still fail if multiple updates occur in the same transaction context, as firstRun would reset with each trigger invocation.

Verdict: Incorrect due to compilation errors and ineffective recursion control.

Option C:
Analysis: This version declares firstRun as a static Boolean initialized to true but does not use it in the handle method to control execution. The handle method always increments Edit_Count__c for each case, with no recursion prevention. Like Option A, this risks multiple increments if the trigger fires again due to a subsequent update. The firstRun variable is unused, making this implementation incomplete.
Verdict: Incorrect due to lack of recursion control.

Option D:
Analysis: Here, the firstRun Boolean is defined within the trigger context, set to true initially. The if (firstRun) condition ensures the handle method runs only once per trigger execution, and firstRun is set to false afterward. Since it’s a local variable in the trigger, it resets with each new transaction but persists within a single trigger invocation. This prevents recursion within the same update operation—e.g., if a workflow updates the record, the trigger won’t re-execute the increment logic. The handle method then safely increments Edit_Count__c for each case. This approach effectively limits the increment to once per update.
Verdict: Correct, as it prevents recursion and ensures a single increment per change.

Conclusion:
The requirement is to increment Edit_Count__c once per Case update, avoiding multiple increments due to recursion. Option D uses a local firstRun Boolean in the trigger to control execution, ensuring the handle method runs only once per update. Options A and C lack recursion control, while Option B has a compilation error due to the instance variable mismatch. Thus, the correct answer is:
D. Option D