I suspect it means that ##f(x_0) = 0## for some specified value of ##x_0## with ##x_0 \neq -2,4##. Then you would, indeed, have three (linear) equations in the three unknowns a,b,c, and you ought to be able to write the solution.think there is a typo in the question. Where it says "f(x) = 0", x should be a number, the same as in f(4) = 0 and f(-2) = -6.
You did the right thing setting up two equations in a, b, and c. If the question was printed correctly you would be able to get three equations.
If f(x) = 0 for every value of x, then a = b = c = 0, but that doesn't make any sense when the question says f(-2) = -6.