Imagining the plane as a checkerboard, there are two exhaustive cases for generalized trominos:
(1) All monominos are the same color. (2) Exactly two monominos are the same color.$\\\ \\\ $
Case (1) can split into two exhaustive sub-cases.
(1.1) Two pairs of monominos are an equal distance apart either horizontally and/or vertically.
(1.2) Three pairs of monominos are an unequal distance apart both horizontally and vertically.
- user69547's answer demonstrates these generalized trominos can tile the plane employing both translation and reflection.$\\\ \\\ $
Case (2) can split into three exhaustive sub-cases.
(2.1) No pairs of monominos are collinear.
(2.2) One or two distinct pairs of monominos are collinear.
These generalized trominos can always tile the plane using only translation by recursively generating infinite parallel lines at an equal distance from one another.
- Copy translates horizontally to the right until doing so will cause a collision.
- Copy the result and join the left-side of the bottom line to the right-side of the top line an infinite number of times.
- Copy the result and join the infinite number of line segments together horizontally an infinite number of times.
- Copy translates of all infinite horizontal lines vertically until no gaps remain.
Step 1 and the first iteration of step 2 are depicted above.
(2.3) Three distinct pairs of monominos are collinear.
- user69547's answer demonstrates these generalized trominos can tile the plane employing both translation and reflection.