Can't argue with the math.
As for me, I was approaching it this way:
We see 2V going through a 2 ohm load in the middle branch. Current is 1 A. Power is 2W.
Now the stated total power is 10W. That means 8 W must be going through the other 2 ohm load at top. THis means 2A goes through it.
To push 2A through a 2 ohm load implies 4V across it.
Where is the 4V supply? We do not see a 4 V source in the circuit. We only see a 2V source.
So I conjectured something the problem doesn't allude to: 'x' must be hiding a voltage supply.
I was skeptical. Could a circuit work the way this problem claimed to? Or was there a mistake?
I used a simulator. I did discover that there has to be a certain voltage coming out of 'x' in order to make the problem make sense (as well as the solution).
I also found out this about the 2V source at right. Current is not flowing out at its top but into it. Thus it does not work as a battery which is the normal meaning of the symbol if we were looking at a normal schematic. (I'm going by convention that the larger plate at top is the positive terminal. And the convention that schematics usually orient a battery with its positive terminal upward.)
So as it turns out, the 2V battery is really a contrivance within the problem to provide oppositional EMF to whatever is the larger voltage source hidden at 'x'. My simulation proves the battery at right pulls current rather than to supply it. However since the 2V battery is not a resistance, no power is consumed.
It puts a bit of a trick into the problem, doesn't it?
Is this a fair problem to give students? Maybe we'd have to say the instructor has a bit of a sense of humor. (Was this for extra credit, by any chance?)
On the one hand it shows us how to use straightforward calculations as one approach to a problem.
On the other hand it forces us to think outside the box when we notice an anomaly.