A lot of people under 40 would've said 'WOULD OF" instead of "WOULD'VE" because they think would've is would of ! what does "would of" even mean?
It's more prompted by the fact Melvyn Bragg is on the radio here talking about dialects. The "would have" construction seems to be sufficiently prevalent to count as a dialect, so I am curious. There are plenty of others: stood standing (sometimes used in conjunction with banjuleles); while in "I've been doing this while 9am"; usage of through versus through to or from and to; different from, different to, different than; and so on.
Second one is clearer, and quite clearly correct. But I'm too tired to decide if the first one is actually wrong or not. People would understand what you meant, which is the main point of language, after all.
"Would of" is the result of someone hearing "would've" (a contraction of "would have") and then spelling it phonetically without understanding the grammar of what they heard. How many times have we seen people write "walaa" when they should have written "voilĂ " to mean something along the lines of "it is revealed" or "see there" or "it is unveiled".
On second thought, should that be "...wouldn't you also have to have had carrots?" Or maybe "....wouldn't you also have had to have had carrots?" (Or not?)