CURL & Remote Mysql is not working

Hello.

What are the security group's inbound rules configured?
If RDS does not enable public access, access from EC2 to RDS will be private access, so you will not be able to communicate even if you register ElasticIP.
If it is a private connection, please refer to the EC2 private IP address and EC2 security group in the security group in the inbound rule.
https://docs.aws.amazon.com/quicksight/latest/user/vpc-security-groups.html#vpc-inbound-rules

Maybe we need to clarify your question first. Just Assume your scenario is about connecting to RDS from EC2 from the same VPC, correct me if in need. Based on the information you provided, I have summarized the key points of the issue:

1. The cURL function in PHP is not working correctly, but running cURL commands in the console works fine.
2. Unable to access a remote MySQL server from cPanel, resulting in a timeout error.
3. The server is pingable, but the connections still fail.

Possible reasons and troubleshooting steps:

1. security group setting a. ensure the outbound rule allow 3306 port to RDS instance; you can configure a rule that allow access to the security group of rds;
2. PHP environment configuration (Most likely the root cause in this case) a. Verify that the cURL extension is correctly installed and enabled in PHP. b. Check the PHP connection timeout settings.
3. MySQL configuration a. make sure the inbound rule of RDS MySQL allows access from EC2, you can also configure a inbound rule to allow access from the security group of EC2; you can also refer to the following docs to identify the root cause in your case .
4. https://repost.aws/knowledge-center/rds-cannot-connect
5. https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/CHAP_Troubleshooting.html

I hope you found this helpful. If you face any other issues or require further assistance, you could also reach out to AWS Support along with your use case details, and we would be happy to assist you further. Thank you!