'a' makes the difference

If $f_a(X) = X^4 - 4X^3 - 8X^2 + a$ then $f_a'(X) = 4X(X-4)(X+1)$ , and so $f_a(X)$ has turning points at $X=-1,0,4$ . Since $f_a(-1) = a-3$ , $f_a(0) = a$ and $f_a(4) = a-128$ , we see that all roots of $f_a(X)$ are real precisely when $0 \le a \le 3$ . SInce we are only interested in integer roots, this makes the answer $0+1+2+3 = \boxed{6}$ .