Let y=ax^3+bx^2+cx+d and plug in the coordinates (x,y):
A: -40=-64000a+1600b-40c+d
B: 4=-8000a+400b-20c+d
C: 4=-1000a+100b-10c+d
D: 68=8000a+400b+20c+d
E: B+D: 72=800b+2d, 36=400b+d, d=36-400b
F: B-C: 0=-7000a+300b-10c; 10c=-7000a+300b, c=-700a+30b=10(-70a+3b)
G: D+2C: 76=6000a+600b+3d=6000a+600b+108-1200b=6000a-600b+108; -32=6000a-600b, -4=750a-75b from which a=(75b-4)/750. But c=-700a+30b=-(700/750)(75b-4)+30b=-40b+56/15=(56-600b)/15.
Now we have a, c and d in terms of b.
Substitute in A: -40=-(64000/750)(75b-4)+1600b-(40/15)(56-600b)+36-400b=
-(256/3)(75b-4)+1600b-(8/3)(56-600b)+36-400b=
-6400b+1024/3+1600b-448/3+1600b+36-400b=
-3600b+228. 3600b=268, b=67/900. So a=19/9000, c=34/45, d=56/9.
y=19x^3/9000+67x^2/900+34x/45+56/9 fits the four points, as can be seen in the graph below. The horizontal coloured lines show the y value at -40, 4 and 68.
