I think I've got this. In cell B1 I put in the formula:
...and formatted it as a number (no decimal places)
on cell B1 I also used Conditional formatting where I selected "Use formula to determine which cells to format" and entered the formula:
...and gave it the format of a red fill
With these settings, if the date entered in A1 is before today plus 8 days (i.e. for every day that passes, the negative number will increase), it will result in a negative number, and the conditional formatting will turn it red.
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