Is this perfectly correct English?
I just think he should have thought about it before he let her move in, because it certainly doesn't look like their relationship is going well at the moment.
Yes, that's correct. In formal English "look as if + clause" should be used instead of "look like + clause", but in your sort of everyday sentence "look like" is OK.
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Yes, that's correct.
In formal English "look as if + clause" should be used instead of "look like + clause", but in your sort of everyday sentence "look like" is OK.