Precalculus, Chapter 9, 9.4, Section 9.4, Problem 69

You need to remember what a quadratic model is, such that:
a_n = f(n) = a*n^2 + b*n + c
The problem provides the following information, such that:
a_0 = 3 => f(0) = a*0^2 + b*0 + c => c = 3
a_1 = 3 => f(1) = a*1^2 + b*1 + c => a + b + c = 3
a_4 = 15 => f(4) = a*4^2 + b*4 + c => 16a + 4b + c = 15
You need to replace 3 for c in equation a + b + c = 3 :
a + b + 3 = 3 => a + b = 0
You need to replace 3 for c in equation 16a + 4b + c = 15 :
16a + 4b + 3 = 15 => 16a + 4b = 12 => 4a + b = 3
Subtract a + b = 0 from 4a + b = 3 , such that:
4a + b - a - b = 3
3a = 3=> a = 1
Replace 1 for a in equation a + b = 0 such that:
1 + b = 0 => b = -1
Hence, the quadratic model for the given sequence is a_n = n^2 - n + 3.