Given load resistance with max power, how to find specific resistance within source
You have to ask yourself; "Is my teacher a mathematician, or an engineer?"


An engineer, like the other answer by Jan, says, "the best I can do is zero ohms", because an engineer knows that negative resistors are hard to come by.
A mathematician is happy with negative answers, because in this case, you have a nice unique solution for both (a) and (b) resistor values.


Infinite power is available for R4 with a negative resistor of the correct value at R1. You solve the total positive resistance to the right of R1, and then make R1 negative so that the 100V source sees a total resistance of zero-ohms. An engineer knows that this solution is not practical. But an engineer also knows that negative resistance is a useful tool - it can be fabricated with some active circuitry, and it also exists in some devices. 

Power for any negative resistance has to come from somewhere. For an active negative resistor, it comes from its own power supply. For a natural negative resistance of a device, it comes from its bias supply. This would apply to discharge tubes, Gunn diode, Esaki diodes...without external power, negative resistance is latent.