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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?

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