She needs to score at least a 97 to get a score of at least 90.
Mean = 83 + 88 + 92 + 97 = 360
360/4 = 90
You could have found this answer without using trial and error.
If she wants to get a mean of 90 and she will have 4 scores, you can multiply these together.
This equals 360.
Now you need to find the minimum score she needs to get. Take 360 and subtract what the original 3 numbers equal, 263.
This gives you 97.
I hope that makes sense :)