It's going to be very hard to get your white background white and still get the exposure correct on your kids. The reason is that to expose for the background so that it is a nice clean white you are going to over expose anything in front of it. Think of the light leaving your flash and traveling across a room. If you have something a foot in front of the flash it is real bright on that object. If you move the object 1 foot futher away but leave the light the same strength (now 2 feet from the flash), the light isn't going to be quit as bright on that object. If you move the object another foot away, again, leaving the flash the same strength (this object will be 3 feet away from the flash) the light falling on it will be even less because the light has spread out as it travels across the room. If you increase the flash so that it is the same brightness at 3 feet as it is at 1 foot from the flash, you have now made the light even brighter at those spots at 2 feet and 1 foot in front of the flash. This would be the same with lighting the background correctly. The subject in front of the background is going to get more light than the background, causing it to be over exposed.
This is why it is very hard to to get a white background and a properly exposed subject with one light. The background will come out gray with one light. You really need two lights, one shines on the background and exposes it for the proper amount and one light shines on the subject and gives the right amount of light for it. You can have light from the subject spill over to the background but you usually do not want light from the background to spill back onto your subject.
What most people do that only have one light is expose for the subject and make the background white in Photoshop. Which is not that easy to do and make it look natural either.
You could mix sunlight and your flash together, the flash lighting the background and the sunlight coming through a window lighting your subject, but that can take a little bit of practice to get them working together as well. You can do it with your camera, but don't get discouraged if it doesn't come out like your examples did. Those were probably taken with 2 or 3 lights.
Mike