We are given the graph of f(x) -- it looks like a parabola (but may not be one) with vertex at (4,-3) opening down.
We are given g(x)=-f(x+3)-3. Note that the graph of g(x) is the graph of f(x) translated 3 units to the left, reflected over the x-axis, and then translated 3 units down. Thus the point (4,-3) is mapped to the point (1,0) with the rest of the graph opening up.
So g(x) has one x-intercept.
As f(x) is a function (given), f(x) has 1 y-intercept.
So a=b=1 and a+b=2.
The graph and its transformation:
** Suppose that the function f(x) was f(x)=-(x-4)^2-3.