In my opinion the question is awful. I have to make so many assumptions. Now, what the heck is that drawing? Are we supposed to assume that is the xy axis without it being labelled? Am I supposed to assume that graph at the ends is touching the x axis, making A amd B roots.
Did they expect you to be able to get the roots to that quartic? Not an easy job without a computer. Do they want the straight line distance between A and B or the Arc length distance?
Now assuming that is the xy axis and A and B are touching it and that squiggly mess is the graph of the function, which it is not. I solve like this:
-X^4+5x^3+4X^2+6X+8 = 0 has 2 real roots thay are:
x = -1 and x = 5.89102041
So the straight line distance is 6.89102041 which is none of your choices.
The arc length distance between A and B is 327.039 also not one of your choices.