Well, that's a sharp observation of yours! What happens is that your book depicts the NUCLEAR reaction, the main realm of nuclear physics, and believe it or not, electrons, I mean, the electrons that orbit around the nucleus (because there can be electrons coming from within the nucleus, as in beta decay), are of no great concern to nuclear processes. That's why you most probably see the alpha particle as superscript 4, subscript 2 He, without indication of the 2+ charge. But we know well that charge is conserved in all processes observed so far, so if you want to be really rigorous, you can represent your alpha particle as a 2+ ion and the superscript 188 subscript 84 Po as a 2- ion (well, that is a likely oxidation state for polonium, anyways).

When you study nuclear reactions in general, there is little information or concern about the chemical state of the radioactive isotopes (that means about the external orbiting electrons of these atoms), because they practically have no influence on the nuclear behavior.

By the way, I am right now doing some experiments in which I use resonant nuclear gamma absorption to study the chemical state of the absorbing nucleus, this is called Moess Bauer spectrometer, and I should say this is the exception among all techniques that deal with nuclear processes, for the great amount of chemical information(again, about electrons in outer shells) it gives.

Keep those discerning eyes wide open!

This is a great question! When you write CHEMICAL equations for the creation of compounds from elements or the dissociation of compounds into elements, you are writing equations that are totally based on the sharing and exchanging of electrons. These reactions always must obey the laws of conservation of mass and conservation of electric charge.

But when you write NUCLEAR equations, involving the particles the nucleus(protons, neutrons, and their constituent quarks), these reactions have to do with instabilities within the nucleus itself, and don't involve the electrons in the electron cloud. Nuclear reactions must obey the conservation of mass and electric charge, but also must obey other conservation laws: conservation of baryon number, lepton number, and still others when you write equations that govern the particles inside the particles (quarks).

Alpha particles, which as you correctly state are helium nuclei with 2 protons and 2 neutrons don't simply "hang out" as helium nuclei inside a nucleus of Radon, for example. The nucleons are always jiggling around and regrouping. When a He grouping happens to be near the surface of the nucleus, there is a certain probability that it will repel from the nucleus.

Now - what happens if you leave a substance that decays by alpha decay in a sealed box is that, after some time, you will find neutral helium gas inside the box! How did that neutral helium gas form? When the alpha particles were ejected from the nucleus they had to go through the electron cloud and they naturally picked up a -2 charge from the electron "cloud" to balance their +2 charge, so that later when you test the air inside the box you find an appropriate amount of neutral helium gas.

Your book did not write this in their nuclear equations, because the electrons from the electron cloud are not directly involved in the nuclear reaction that converts 86-Ra-222 to 84-Po-220, as that is a nuclear reaction. But, that is what becomes of the extra electrons - AFTER the alpha particle leaves the nucleus it attracts the -2 charge from the electron cloud to make neutral helium gas.

Generally, radioactive decay energies are far in excess of those needed for electron ionization of small nucleus such as He. You are correct that the electrons no longer balance and so two of them will depart the originator, however, the alpha particle will almost certainly not take them as it will be scattering with other atoms on is way out -- and such interactions would rapidly ionize it anyway. There is another possibility, however, which is electron capture in the nucleus, sort of beta decay running backwards to make a proton into a neutron.

I hope this short answer help.

I think one way to look at this is to ask what happens to the electrons when a diatomic molecule, like H₂, N₂, or O₂ breaks apart. For simplicity, lets say that a H₂ molecule breaks up into two hydrogen atoms. So there are two positively charged protons moving away from each other with two electrons floating around. Do you think the two electrons would stay with one of the protons or would one electron go with each proton? Why? So what do you think would (eventually) happen to the two extra electrons that are around the Polonium nucleus once the alpha particle shoots off?