Substitute the x and y values for each point:
6=a-b+c; 4=a+b+c; 9=4a+2b+c.
Add the first two equations: 2a+2c=10, so a+c=5.
Subtract twice the second equation from the third: 2a-c=1. Now we have two new equations in a and c. Add them together: 3a=6 so a=2 and c=5-2=3. Therefore b=4-(a+c)=4-5=-1.
Therefore the quadratic is 2x^2-x+3. Check that the three points satisfy the equation:
x=-1: y=2+1+3=6; x=1: y=2-1+3=4; x=2: y=8-2+3=9.