add examples to use the gateway aside from the shorthand

docs/gateway.md

@@ -35,6 +35,40 @@ from pydantic_ai import Agent
 
 agent = Agent('gateway/openai:gpt-5')
 
+result = agent.run_sync('Where does "hello world" come from?')
+print(result.output)
+"""
+The first known use of "hello, world" was in a 1974 textbook about the C programming language.
+"""
+```
+```python {title="passing_api_key.py"}
+from pydantic_ai import Agent
+
+from pydantic_ai import Agent
+from pydantic_ai.providers.gateway import gateway_provider
+from pydantic_ai.models.openai import OpenAIChatModel
+
+provider = gateway_provider('openai', api_key='PYDANTIC_AI_GATEWAY_API_KEY')
+model = OpenAIChatModel('gpt-5', provider=provider)
+agent = Agent(model)
+
+result = agent.run_sync('Where does "hello world" come from?')
+print(result.output)
+"""
+The first known use of "hello, world" was in a 1974 textbook about the C programming language.
+"""
+```
+```python {title="routing_via_provider.py"}
+from pydantic_ai import Agent
+
+from pydantic_ai import Agent
+from pydantic_ai.providers.gateway import gateway_provider
+from pydantic_ai.models.openai import OpenAIChatModel
+
+provider = gateway_provider('openai', api_key='PYDANTIC_AI_GATEWAY_API_KEY')
+model = OpenAIChatModel('gpt-5', provider=provider, route='modal-ai')
+agent = Agent(model)
+
 result = agent.run_sync('Where does "hello world" come from?')
 print(result.output)
 """