Make sure the circles are aligned either vertically or horizontally, then draw a rectangle between them, ensuring that the size matches and that it ends at the centers of both circles. Then merge with the pathfinder tool and rotate/recolor as needed.
The case is already solved with a tangent drawing extension. Actually in this special case the tangents can be drawn also without that extension. They are as long as the line between the circle midpoints and they can be made by duplicating that connection line. Placing them properly with snapping happens when one draws a 90 degrees rotated copy of the line. Its midpoint snaps to the circle midpoint. In inkscape snap to crossing works well.
The tangents are joined, the resulted rectangle and the circles are combined with Path > Union.
BTW in Inkscape closed areas like circles often cause false snaps if they have a stroke. For that reason it's useful keep them strokeless during the construction.
If it is acceptable to place the two circles either vertically or horizontally to perform the creation, then rotate the creation, the process is straightforward.
Use the align feature for placement. Convert the circles to paths (Path, Object to Path) and break the nodes for both circles at the diameter. Delete the semi-circles that are no longer needed. Add lines between the remaining semi-circles. Join the nodes and fill with desired color.