What Is The Balanced Equation For Hydrochloric Acid And Ammonium Hydroxide?

Ammonium Hydroxide? That is just ammonia in water. Get a NH4+ and OH- and obviously the H+ will be attracted to the OH- and you'll get water. If you have some other source of OH-(NaOH) then the vast majority of the ammonia will be NH3. Apart from that, yeh:
NH3+HCl -->NH4Cl