Plug in the (x,y) values given:
(1) 3=a+b+c
(2) -1=9a+3b+c
(3) 0=16a+4b+c
We have 3 equations and 3 unknowns, so we have a potential solution.
(4): (2)-(1): -4=8a+2b
(3)-(2): 1=7a+b, so b=1-7a. Substitute for b in (4): -4=8a+2-14a=-6a+2; 6a=6, a=1, so b=1-7=-6 and c=3-(a+b)=8.
So y=x^2-6x+8.