diff --git a/starlette/responses.py b/starlette/responses.py
index 507e79b..b33aa86 100644
--- a/starlette/responses.py
+++ b/starlette/responses.py
@@ -311,7 +311,7 @@ class FileResponse(Response):
         content_length = str(stat_result.st_size)
         last_modified = formatdate(stat_result.st_mtime, usegmt=True)
         etag_base = str(stat_result.st_mtime) + "-" + str(stat_result.st_size)
-        etag = md5_hexdigest(etag_base.encode(), usedforsecurity=False)
+        etag = f'"{md5_hexdigest(etag_base.encode(), usedforsecurity=False)}"'
 
         self.headers.setdefault("content-length", content_length)
         self.headers.setdefault("last-modified", last_modified)
diff --git a/starlette/staticfiles.py b/starlette/staticfiles.py
index 30fbc11..895105a 100644
--- a/starlette/staticfiles.py
+++ b/starlette/staticfiles.py
@@ -225,7 +225,7 @@ class StaticFiles:
         try:
             if_none_match = request_headers["if-none-match"]
             etag = response_headers["etag"]
-            if if_none_match == etag:
+            if etag in [tag.strip(" W/") for tag in if_none_match.split(",")]:
                 return True
         except KeyError:
             pass

diff --git a/tests/test_staticfiles.py b/tests/test_staticfiles.py
index 23d0b57..2bb7305 100644
--- a/tests/test_staticfiles.py
+++ b/tests/test_staticfiles.py
@@ -208,6 +208,11 @@ def test_staticfiles_304_with_etag_match(tmpdir, test_client_factory):
     second_resp = client.get("/example.txt", headers={"if-none-match": last_etag})
     assert second_resp.status_code == 304
     assert second_resp.content == b""
+    second_resp = client.get(
+        "/example.txt", headers={"if-none-match": f'W/{last_etag}, "123"'}
+    )
+    assert second_resp.status_code == 304
+    assert second_resp.content == b""
 
 
 def test_staticfiles_304_with_last_modified_compare_last_req(

Strong and weak etags for StaticFiles
### Discussed in https://github.com/encode/starlette/discussions/2297

<div type='discussions-op-text'>

<sup>Originally posted by **markus1978** October  6, 2023</sup>
tl;tr: It seems that `starlette.staticfiles.StaticFiles` does not use "proper" HTTP etags which causes issues with reverse proxies and browsers.

The `starlette.staticfiles.StaticFiles` class supports etags, e.g. it creates a `Etag` header in all responses and considers `If-None-Match` headers in its `is_not_modified` method. However, it only puts the plain hash values in the response `Etag` header and also only interprets `If-None-Match` values as plain hash values . HTTP Etags are either weak or strong and the values are supposed to be wrapped in `W/"..."` or `"..."` respectively (https://datatracker.ietf.org/doc/html/rfc7232#section-2.3.1).

This causes two problems for me. First, nginx as a reverse proxy is considering all starlette etag headers as weak etags and the gzip module removes. Secondly, when the browser sends a strong or weak etag in the `If-None-Match` header, it never matches because `'...some-tag...' != '"...some-tag..."'`. 

I currently use this work arround  to solve my issues:
```python
class StaticFiles(StarletteStaticFiles):

    etag_re = r'^(W/)?"?([^"]*)"?$'

    def is_not_modified(
        self, response_headers: Headers, request_headers: Headers
    ) -> bool:
        try:
            if_none_match = request_headers["if-none-match"]
            # I remove all weak/strong etag syntax, basically doing a weak match 
            match = re.match(StaticFiles.etag_re, if_none_match)
            if_none_match = match.group(2)
            etag = response_headers["etag"]
            if if_none_match == etag:
                return True
        except KeyError:
            pass

        return super().is_not_modified(response_headers, request_headers)

    def file_response(
        self, full_path: PathLike, stat_result: os.stat_result, scope: Scope, 
        status_code: int = 200,
    ) -> Response:
        response = super().file_response(full_path, stat_result, scope, status_code)
        # I add strong etag syntax, basically wrapping the etag value in "..."
        if 'etag' in response.headers:
            response.headers['etag'] = f'"{response.headers.get("etag")}"'
        return response
```

I am not an expert in etags, just trying to make it work in our app. I am glad for any hints, if i am getting it wrong or missing anything here.

For reference, the relevant part from the current starlette.
From `staticfiles.py::StaticFiles`
```python
    def is_not_modified(
        self, response_headers: Headers, request_headers: Headers
    ) -> bool:
        """
        Given the request and response headers, return `True` if an HTTP
        "Not Modified" response could be returned instead.
        """
        try:
            if_none_match = request_headers["if-none-match"]
            etag = response_headers["etag"]
            if if_none_match == etag:
                return True
        except KeyError:
            pass

        try:
            if_modified_since = parsedate(request_headers["if-modified-since"])
            last_modified = parsedate(response_headers["last-modified"])
            if (
                if_modified_since is not None
                and last_modified is not None
                and if_modified_since >= last_modified
            ):
                return True
        except KeyError:
            pass

        return False
```

From `responses.py::FileResponse`
```python
    def set_stat_headers(self, stat_result: os.stat_result) -> None:
        content_length = str(stat_result.st_size)
        last_modified = formatdate(stat_result.st_mtime, usegmt=True)
        etag_base = str(stat_result.st_mtime) + "-" + str(stat_result.st_size)
        etag = md5_hexdigest(etag_base.encode(), usedforsecurity=False)

        self.headers.setdefault("content-length", content_length)
        self.headers.setdefault("last-modified", last_modified)
        self.headers.setdefault("etag", etag)
```</div>

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tests/test_staticfiles.py::test_staticfiles_304_with_etag_match[asyncio]
tests/test_staticfiles.py::test_staticfiles[asyncio] tests/test_staticfiles.py::test_staticfiles_with_pathlib[asyncio] tests/test_staticfiles.py::test_staticfiles_head_with_middleware[asyncio] tests/test_staticfiles.py::test_staticfiles_with_package[asyncio] tests/test_staticfiles.py::test_staticfiles_post[asyncio] tests/test_staticfiles.py::test_staticfiles_with_directory_returns_404[asyncio] tests/test_staticfiles.py::test_staticfiles_with_missing_file_returns_404[asyncio] tests/test_staticfiles.py::test_staticfiles_configured_with_missing_directory[asyncio] tests/test_staticfiles.py::test_staticfiles_configured_with_file_instead_of_directory[asyncio] tests/test_staticfiles.py::test_staticfiles_config_check_occurs_only_once[asyncio] tests/test_staticfiles.py::test_staticfiles_never_read_file_for_head_method[asyncio] tests/test_staticfiles.py::test_staticfiles_304_with_last_modified_compare_last_req[asyncio] tests/test_staticfiles.py::test_staticfiles_html_normal[asyncio] tests/test_staticfiles.py::test_staticfiles_html_without_index[asyncio] tests/test_staticfiles.py::test_staticfiles_html_without_404[asyncio] tests/test_staticfiles.py::test_staticfiles_html_only_files[asyncio] tests/test_staticfiles.py::test_staticfiles_cache_invalidation_for_deleted_file_html_mode[asyncio] tests/test_staticfiles.py::test_staticfiles_with_missing_dir_returns_404[asyncio] tests/test_staticfiles.py::test_staticfiles_access_file_as_dir_returns_404[asyncio] tests/test_staticfiles.py::test_staticfiles_unhandled_os_error_returns_500[asyncio] tests/test_staticfiles.py::test_staticfiles_follows_symlinks[asyncio] tests/test_staticfiles.py::test_staticfiles_follows_symlink_directories[asyncio] tests/test_staticfiles.py::test_staticfiles_instantiated_with_missing_directory tests/test_staticfiles.py::test_staticfiles_prevents_breaking_out_of_directory tests/test_staticfiles.py::test_staticfiles_disallows_path_traversal_with_symlinks tests/test_staticfiles.py::test_staticfiles_avoids_path_traversal