If it still bleeds after two go-rounds with Retayne, it's probably better used for something else. One thing to be aware of; not all whites will pick up color. Try dampening a piece of the white fabric you will be using and rub it against the red. If it doesn't pick up color, the whole bleeding thing may be a non-issue. To make sure you could throw in a piece of the white fabric and wash it with the red. (I would do that after the Retayne to make sure that there will be no color transfer later.)