Please see the attached image.
My brain is refusing to work right now, I don’t want to do use a boolean modifier.
The fastest way is to make a plane, and then depending on the topology you need you can extrude and scale down a hole, then subdivide and use loop tools to make a circle, or to just start with subdivisions and do the same thing. Here is a blend file with both types. Hope it helps!
create a cube, scale it to a rectangle.
Select the bottom lower face and delete only the face, not the edge-vertices!
Select the front and back face and delete it too
create a tube without caps, delete half of the tube.
if you did both as separate objects, you have to join them into one object.
(if you did it in edit-mode, the meshs are already one object)
move the half-cut tube to the lower positione, where you deleted the face.
connect the edge of the half-cut tube on the left and right side with the left/right
side of the scaled-cube-sides and create the faces.
for quick face-fill, select all front vertices and fill (Alt-F) and
same with back vertices.
Tipp: always try to find simple base meshes to cut, scale and combine to get the final object.
Forgot it: because the tube(zylinder) is cut it is necessary to switch the face normals or this part of the object or we will have face-normals in the wrong direction for this part.
I’d go for a different technique
start with a circle(12 vert no fill)
in edit mode delete the bottom half of the circle
add in a plane and scale it up 3x . S, 3, enter
make sure the bottom is aligned with the bottom vertices (center of circle)
add three edge loops to the plane vertically (ctrl+R and scroll up twice)
add two edge loops either side of the center edge loop
add two edge loops horizontally
remove the vertices that are inside the circle
and patch it up
then just extrude for depth
see reference image
hope this helps 
Ahhh perfect thanks everyone.
I just realized how awful my explanation was. I hadn’t slept for like 36 hours when I wrote it haha. I didn’t even mention deleting the lower half of the plane after subdividing. Anyway, as with most shapes like this, it is easiest to make a 2D representation of it and then just extrude it to give it depth. And in this case, the loop tools addon makes it an absolute breeze.