An Opportunity needs to have an amount rolled up from a custom object that is not in a master- detail relationship. How can this be achieved?

Standard roll-up summary fields only function across master-detail relationships. To replicate this functionality for a lookup relationship, an Apex trigger is the appropriate solution. The trigger should be placed on the child object. When a child record is created, updated, deleted, or undeleted, the trigger fires. Inside the trigger, an aggregate SOQL query (e.g., SELECT SUM(Amountc) FROM Childc WHERE Opportunityc = :oppId) can calculate the new total for the parent Opportunity and update a custom field on it.

A. A trigger on the parent (Opportunity) object will not fire when a child record is modified, so it cannot react to changes that require a roll-up recalculation.

B. The Streaming API is a notification service for pushing data changes to external clients; it is not used for server-side data aggregation.

D. The Metadata API is used for deploying and retrieving an org's customizations (metadata), not for performing data-level calculations.

Salesforce Apex Developer Guide, "Triggers". "Triggers can be used to perform custom actions before or after changes to Salesforce records, such as insertions, updates, or deletions."

Salesforce Apex Developer Guide, "Working with SOQL Aggregate Functions". "Use aggregate functions in a GROUP BY clause in SOQL queries to generate reports for analysis... Common functions include SUM() and COUNT()." This is the mechanism used within the trigger.

Trailhead by Salesforce, "Apex Triggers" module, "Trigger Apex". This module explains how triggers respond to DML events, which is the foundation for this roll-up pattern.