Thank you for the opportunity to help you with your question!
When graphing a polynomial, all of the roots of the polynomial are the points at which the graph intersects the x-axis. If the root is an odd degree root (single root, triple root, 5th degree root, etcc...) then the graph of the function will cross through the x-axis. If the root is an even degree (double, quadruple, etc...) then the graph of the function will touch the x-axis but not actually pass through. Unfortunately, there is no easy way to tell a double root from any other even degree root when looking at the graph.
So, for the short answer, any place where the graph of the polynomial touches the x-axis but does not pass through is a potential double root.
Please let me know if you need any clarification. I'm always happy to answer your questions.