When a voltage is applied across two resistors in series the voltage across each of the resistors is not the same. It differs as the resistance is different. On the other hand, the current passing through both the resistors is the same. This has to be the case as the current that starts to flow from one end is not absorbed on its way, all the current has to reach the other end.
The power produced in both the resistors can be found from the electric current flowing them. P = I^2*R, we need to determine the equivalent resistance of the two resistors to determine the current. For two resistors with a resistance R1 and R2 in series, the equivalent resistance is R1 + R2.
Here, the equivalent resistance is 50 + 300 = 350 ohm. As a voltage of 50 V is applied across them, the current is 50/350 = 1/7 A.
When 1/7 A flows through the 50 ohm resistor the power produced is (1/7)^2*50 = 1.02 W. When 1/7 A flows through the 300 ohm resistor, the power produced is 6.12 W.
The power produced by the 50 ohm resistor is 1.02 W and the power produced by the 300 ohm resistor is 6.12 W.