chess online

In what regard are you asking the question?

After 3.Rg8 white has the threat of Bg7 threatening to skewer the queen, which must be dealt with. 3....Ke7 is no good for black because after 4.Bf8+ black is under constant attack and near as I can tell loses both the rook and bishop withloss of tempo. Moving the black bishop accomplishes absolutely nothing, so that line can be quickly discounted. Moving blacks rook doesn't seem useful either, It has no good defensive position to take up and Rc1+ is meaningless as it loses the rook. a6 or a5 is pointless as well.

Which brings attention to the black queen. The black queen cannot move of the a1-h8 diagonal without dropping the rook and allowing white to initiate an attack with tempo, thus if the queen is to move it must check the white king. By process of elimination, the only move black really has is 3....Qh2+, which is what's given.

I'm hesitant still because of 3.Rg8 Ke7 4.Bf8+ Kd7 line, which isn't examined in the solution and appears highly tactical for both sides.

4...Kg6 gives Qh6#
4...Ke8 or d8 leads to 5.Bxd6+, exposed check and loss of the bishop and queen
4...Kd7 looks bad for black after 5.Bxd6+ Qxd6 and either of 6.Rd8+ or 6.Qxc3

I have to agree with the solution as given, if only for the reason black can still be annoying.