Aluminum hydroxide reacts with an excess of hydroxide ions to form the complex ion Al(OH)4-

a)Write an equation for this reaction
b) Calculate K.
c) Determine the solubility of Al(OH)3 in (mol/L) at pH 12.0

My equation was
Al(OH)3 + OH- --> Al(OH)4-
but I'm wondering if I may need to balance out the OH- with H+?
Also, I'm a little confused on what K is. I know it's an equilibrium constant, but how does it differ from Ksp or Kc or Kp? Do you have to decide which constant you're looking for? And how do I find this once I determine the right constant?

Your equation is correct for the formation of Al(OH)4^-. There is nothing to balance out; i.e., atoms balance, charge balances, so nothing needs to be added.
The formation constant is just another equilibrium constant but it pertains to the formation of complex ions; therefore, they have given the name of formation constant to it just as they gave solubility constant to the solubility of ppts. The K expression is the same as all the others; that is, (products)/(reactants) with each raised to a power indicated by the coefficient.
I don't know how to calculate K with "nothing" given.
If you find K, however, then
K = [(Al(OH)4^-)/(OH^-)]
Plug in OH^- and knowing K calculate (Al(OH)4^-.

I made a typo above.
....to it just as they gave solubility constant...... should read
....to it just as they gave solubility PRODUCT constant.......

To write the equation for the reaction, you have already correctly written it as:

Al(OH)3 + OH- → Al(OH)4-

There is no need to balance the hydroxide ions with H+ because the reaction is occurring in a basic solution, and hydroxide ions are already present as OH-.

Now, let's discuss the different equilibrium constants:

1. Ksp (Solubility Product Constant): This constant is used for sparingly soluble salts and represents the equilibrium between the dissolved ions and the solid salt. It is calculated using the concentrations (or solubilities) of the ions in the solution.

2. Kc (Equilibrium Constant in terms of concentrations): This constant is used for reactions in a homogeneous phase (i.e., all reactants and products are in the same phase) and is calculated using the concentrations of the species involved.

3. Kp (Equilibrium Constant in terms of partial pressures): This constant is used for reactions involving gases. It is calculated using the partial pressures of the gases involved.

In this case, since we are dealing with a complex ion formation reaction in a basic solution, we will be using the formation constant, denoted as Kf or simply K. The formation constant represents the equilibrium between the complex ion and its constituent ions.

Unfortunately, you have not provided any specific values for concentrations or any other information needed to calculate K. Therefore, without more information, it is not possible to determine the numerical value of K or the solubility of Al(OH)3 at pH 12.0.

If you have the necessary information, such as the initial concentrations or a given value for K, I can help you further calculate the solubility of Al(OH)3 at pH 12.0.