In my Android project, I am not quite sure how to make my background image fill the entirety of the
<?xml version="1.0" encoding="utf-8"?>
<!-- Other elements -->
Other than turning your image into a nine patch I don't think this is possible. What you could do instead is add an ImageView as the first view in your RelativeLayout. Set layout_centerInParent to true and have the layout_width and layout_height set to match_parent. Then set scaleType to centerCrop. That will make sure the image fills the screen without any distortion, but depending on screen size/orientation either some of the top/bottom or left/right of the image may be cut off.
<ImageView android:layout_width="match_parent" android:layout_height="match_parent" android:layout_centerInParent="true" android:scaleType="centerCrop" android:src="@drawable/background" />
Any other views in the RelativeLayout will appear on top of the ImageView, as long as it is the first view in the RelativeLayout (when you are in the xml).