Is Qxe8 reasonable?
I always wanted to trade off a queen against two rooks and the engine likes this idea, but am I leaving my pawns unprotected in this situation?
It would be hard to protect those with rooks.
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This is a losing position for White. Normally, trading for a queen against two rooks is reasonable but Black has an extra knight here...
The best objective move sequence would be:
1.Qxe8 Rxe8 2.Rxe8+ Kf7 3.Re2 Qb6+
White's position after 3...Qb6+ is absolutely hopeless - the rooks have no useful squares. White has no counterplay, no attacks, no chance for Black to make a mistake. Very easy win for Black.
If this were a GM game, the White player would have resigned. Here, your best swindle move is 1.Qd3. By keeping the queen on the board, you --might-- be able to generate a miracle comeback.
A possible line:
1...Nxf4?? 2.Rxe8! Nxd3 3.Rexf8# checkmate!
The above line is not impossible in a blitz game. But it wouldn't have happened if you played 1.Qxe8.
Don't follow the computer analysis. Computer lines are meaningless in a lost position. You will need to try a human move. Don't trade your queen for the rooks. You need it to give troubles for the Black players.
In this particular position Qxe8 is the best possible chance even though white is still in a bad shape.
White is down a piece and any other queen move(Qf3/d4/d3) would simply drop f4 pawn after Rxe1 and N(R)xf4 and leave no counterplay for white.
The general rule of thumb is that 2 rooks are worth a queen and a pawn in an endgame.
Queen can be worth more if rooks are disconnected, while two rooks can be worth more if queen's pawns are isolated. Two rooks can support a single advanced pawn rather well.
Once again, generally you'd be happy to give up a queen for two rooks provided you are not dropping anything else within the next few moves.
Vice versa if have a chance to win a queen for two rooks, you better make sure your queen can wreck some extra damage.
In order to get a good assessment of a position, you should consider as many factors as there are: material, tactical, positional, dynamical, permanent, structural, technical, king's safety, etc.
These factors have different weight. And in the given position, the clearly dominant feature is the material disequilibrium (as usual): black is a full knight ahead and such a big material advantage is decisive. Some other minor considerations have little weight.
But in general, two Rooks are better than a Queen, unless the rooks coordinate poorly.
So, in conclusion: White is lost because is a piece down, and probably the best move it has is Qxe8.