Final answer:
To factor the expression 12x^2 + 5x - 12, we find that (3x + 4)(4x - 3) is the correct factorization after splitting the middle term and using the grouping method. There appears to be a typo in the options provided as they all contain 'n' instead of 'x'.
Step-by-step explanation:
To factor the expression 12x2 + 5x - 12, we need to find two numbers that multiply to give the product of the coefficient of x2 (12) and the constant term (-12), which is -144, and also add up to the coefficient of the x term (5). After some trial and error, we find that 16 and -9 meet these criteria. Therefore, we can rewrite the middle term (5x) as 16x - 9x, and the expression becomes:
12x2 + 16x - 9x - 12
Now, we group the terms:
(12x2 + 16x) - (9x + 12)
Factor by grouping:
4x(3x + 4) - 3(3x + 4)
Finally, factor out the common binomial factor (3x + 4):
(3x + 4)(4x - 3)
The correct answer to our factoring challenge is (3x + 4)(4x - 3), which is not listed in the answer choices because they all contain 'n' instead of 'x'. It seems there was a typo in the answer choices provided.