They're asking for the number of integers from 1-100 that aren't the square of an integer, but it's going to be a lot easier and faster to count how many are the square of an integer*:
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100 ...and we're done! (Any higher will be out of the range we care about.)
Count how many we've got. I'm counting 10.
There are 100 integers between 1 and 100 inclusive (inclusive just means "including the ends," so we are counting both 1 and 100), and only 10 of them are squares of integers. Which means 90 aren't. The answer is (E).
* This is kinda like a shaded region area problem in that way...it's easier to look for the opposite of what they ask for, and then subtract.