Write the balanced chemical reaction for the rusting of iron?

the acidic vinegar dissolved a protective coating off the steel wool, allowing the oxidation of the steel to begin. This process is called rusting. In this reaction, iron(III) is combined with oxygen gas to form iron oxide.

Metal iron + oxygen = rust
Fe + O^2  FeO^2
4Fe3+ + 9O2 -> 6Fe2O3

Is this answer correct?

You have two or three things going on here and I don't know what equation you want. If we go with the word equation, then
metal iron + oxygen = rust and
4Fe + 3O2 ==> 2Fe2O3 is the chemical equation.

The question also talks about acidic vinegar dissolving a coating and the iron(III) formed combines with O2 gas to form iron oxide. The equation you have for that is not correct because there are 4 Fe ions on the left and 12 on the right, and the charge on the left is +3 while on the right it is zero. Technically, I think you would need to address the following:
Fe + acid ==> iron(II) ions
Iron(II) ions + oxygen ==> iron(III) + oxide
iron(III) + oxygen ==> Fe2O3.
There is also the matter of water and how it reacts. The rusting of iron, although common place, is a complicated process.
Which equation do you want?

To address the different reactions involved in the rusting of iron, we can break it down into two steps:

1. Dissolution of the protective coating and formation of iron(II) ions:
Metal iron reacts with acidic vinegar (dilute acetic acid) to dissolve any protective coating and form iron(II) ions. The balanced chemical equation for this reaction is:
Fe + 2CH3COOH -> Fe(CH3COO)2 + H2

2. Oxidation of iron(II) to form iron(III) oxide (rust):
Iron(II) ions combine with oxygen gas from the air to form iron(III) oxide. The balanced chemical equation for this reaction is:
4Fe(II) + 3O2 -> 2Fe2O3

So, when we combine both reactions, the overall balanced chemical equation for the rusting of iron in the presence of acid would be:
Fe + 2CH3COOH + 3O2 -> Fe2O3 + 2CH3COOH + H2

It's important to note that the rusting process is not solely limited to these reactions and can involve other factors such as the presence of moisture in the air.