How many moles of P4O10 are produced when we react 8 moles of phosphorous with oxygen? I know the answer is:

4 mol P produces 1 mol P4O10
8 mol P produces 2 mol P4O10.
Or, 1 mol P4O10 x (8 mol P / 4 mol P) = 2 mol P4O10.

but, what is the balanced equation?

I worked this elsewhere for you.

To determine the balanced equation for the reaction between phosphorous (P) and oxygen (O), we can use the following steps:

First, we write the unbalanced equation:
P + O2 → P4O10

Next, we balance the equation by adjusting the coefficients in front of the chemical formulae to ensure that the same number of atoms of each element are present on both sides of the equation. Here's how it can be balanced:

P4 + O2 → P4O10

The balanced equation shows that four moles of phosphorous react with five moles of oxygen to produce ten moles of P4O10.

Therefore, the balanced equation for the reaction between phosphorous and oxygen is:
4P + 5O2 → 2P4O10