Can a 4x11 rectangular grid of square blocks be covered (having no overlaps), with multiple copies of the tetromino (including rotations, but not reflections) as shown below?

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If so, give an example. If not, provide a proof that this cannot be achieved.

__Note__: Each of the 44 square blocks of the 4x11 grid has the same shape and size as each of the 4 square blocks of the tetromino.

I agree that there seems to be no solution. A quibble: the pieces could be better described as "J" shaped (an "L" shape would be a disallowed "reflection"). Two will fit together to form a 2x4 rectangle. A construction would prove the contrary, but I do not know what form an impossibility proof might take.