When the square of three times a positive integer is decreased by the integer, the result is $2010$. what is the integer?

x = a positive integer
3x = 3 times the number
9x^2 = the square of 3 times the number
.
9x^2-x = 2010
.
9x^2 -x -2010 = 0
So x = 15 is the only integer solution.
.
Checking this value, we find:
9*225 -15 = 2010
correct.
.
x = 15 ;)