docs: warn about BackgroundTasks + Response background interaction

Add a warning to the "Return a Response Directly" docs page explaining
that returning a Response with a background parameter silently overrides
any BackgroundTasks added via dependency injection.

Closes #11215

docs/en/docs/advanced/response-directly.md

@@ -74,6 +74,20 @@ When using a `response_model` or return type, FastAPI won't use the `jsonable_en
 
 Instead it takes the JSON bytes generated with Pydantic using the response model (or return type) and returns a `Response` with the right media type for JSON directly (`application/json`).
 
+## Background Tasks and Custom Responses { #background-tasks-and-custom-responses }
+
+/// warning
+
+If you return a `Response` with a `background` parameter **and** also use the `BackgroundTasks` dependency injection, the injected background tasks will be **silently ignored**.
+
+FastAPI only assigns injected background tasks to the response when `response.background is None`. If your `Response` already has a background task attached, the injected tasks are dropped without any error or warning.
+
+Use one mechanism or the other, not both in the same endpoint:
+
+{* ../../docs_src/response_directly/tutorial003.py hl[18,25] *}
+
+///
+
 ## Notes { #notes }
 
 When you return a `Response` directly its data is not validated, converted (serialized), or documented automatically.

docs_src/response_directly/tutorial003.py

@@ -0,0 +1,26 @@
+from fastapi import BackgroundTasks, FastAPI
+from starlette.responses import JSONResponse
+
+app = FastAPI()
+
+
+def write_log(message: str):
+    with open("log.txt", mode="a") as log:
+        log.write(message)
+
+
+# Correct: use only BackgroundTasks (no background= on Response)
+@app.get("/correct")
+async def send_notification(background_tasks: BackgroundTasks):
+    background_tasks.add_task(write_log, "Notification sent")
+    return JSONResponse(content={"message": "done"})
+
+
+# Wrong: background= on Response silently overrides BackgroundTasks
+@app.get("/wrong")
+async def send_notification_wrong(background_tasks: BackgroundTasks):
+    background_tasks.add_task(write_log, "This task will NOT run")
+    return JSONResponse(
+        content={"message": "done"},
+        background=None,  # If this were a BackgroundTask, it would override the line above
+    )