Question 884252
we are dealing with a quadratic sequence and we work with the y values
y | -3 | 0 | 5 | 12 | 21 |
first difference is  3  5  7  9 (subtract first y value from second y value, second y value from third y value, ....)
second difference is 2  2  2  2 
since the constant is 2, we know that we have a n^2 term in the expression for the nth term in the quadratic sequence
again, we look at the y values
y | -3 | 0 | 5 | 12 | 21 |
nth  1   2   3    4    5
n^2  1   4   9   16   25
subtract n^2 from y values
    -4  -4  -4   -4   -4
the nth term in the geometric sequence is n^2 - 4
we can work this from another direction, we know
-3 = a(1^2) +b(1) +c
0 =  a(0^2) +b(0) +c 
5 =  a(5^2) +b(5) +c
or
-3 = a +b +c
0 = c
5 = 25a + 5b +c
we know c = 0
-3 = a +b
1 = 5a +b
this can be solved for a and b
a = 1, b = -4
y = x^2 -4x is the quadratic form