This is strange - on this website, it is not showing a white background. Why is that?
I'm only guessing, but probably you used a Single Scan option, with Trace Bitmap. So that means you only have one path for the result. Since Inkscape's background is transparent by default (even though it looks white) when you export, Inkscape only "sees" the bear path.
If you want a background, you can draw a rectangle, fill it with the color you want, and put behind it.
Hmm, that's strange, I thought I had changed the bear from a raster to a vector with trace bitmap. Then I threw out the raster of the bear.
The reason i threw it out and used the vector is that the forest background image i turned into a pattern would not fill into a raster. It filled the vector bear though.....
Am quite confused now
Lazur is saying that he undid everything you had done, and started all over. So that's why he traced the bear again. In the screenshot you showed, with the error message, I see the bear behind it, with the forest in it. It looks correct to me!
It's hard to say what the problem is with the export. When you first open the Export PNG dialog, first you need to look at the 4 buttons at the top. Inkscape can export 4 different areas -- it can export only what is selected (Selection button), or everything that is drawn on the canvas (Drawing), or just the page (inside the page border) (Page), or a custom area (Custom button). So you need to choose which one you want.
I've never used the custom area before, so I'm not familiar with it. But these are the options I usually use.
If it's just one thing on my canvas that I want to export, I choose Selection. And then make sure it's selected, before clicking Export button. If I want to show my whole drawing, no matter where it is on the canvas, I might choose Drawing. If it happens that I have everything inside the page border, then I use Page.
Or another thing I do sometimes. Usually I have the page border turned off, so that when I'm drawing, I have no idea whether everything is inside the border or not. But there's a quick fix. I select the finished drawing, then Document Properties > Page tab > Resize page to contents. This moves the page border and resizes it to perfectly fit the drawing, all at once. You can also add a margin. Then I use the Page export option.