Is 2×3×4 the prime factorization of 2

No, it is not because prime factorization can only involve prime numbers. 4 is not a prime number but is made up of the prime factors 2x2

so the answer would be: 2x2x2x3 because the 4 will be split into 2x2

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No.
This is because 4 is not prime. You cannot have a prime factorization with composite numbers in it.

The prime factorization should be 2*2*2*3, or 2^3 * 3.