answer is 2.
solution:
Let f(x) = axn, g(x) = - x2 - bx - c. We have to find the number of the real solutions of the equation f(x) = g(x). Since graphs of y = f(x), y = g(x) are intersect at most two different points, maximum number of the real solutions of the equation axn + x2 + bx + c = 0 is 2.