Final answer:
The quadratic expression 3a² + 5a - 12 can be factored by finding two numbers that multiply to 3 times -12 and add up to 5. The correct numbers are 4 and -3, leading to the factored form (3a - 4)(a + 3). This matches option 2 provided in the question.
Step-by-step explanation:
The student has asked to factor the quadratic expression 3a² + 5a - 12 completely. To factor this expression, we need to find two numbers that multiply to give the product of the coefficient of a² (which is 3) and the constant term (which is -12), and at the same time, these two numbers should add up to the coefficient of the a term (which is 5).
The two numbers that meet these criteria are 8 and -3 since (8)(-3) = -24 (which is 3 multiplied by -12) and 8 + (-3) = 5. Now we can express the middle term of the quadratic as a combination of two terms using these numbers: 3a² + 8a - 3a - 12.
Next, we factor by grouping, which involves grouping the terms into pairs that have a common factor: (3a² + 8a) + (-3a - 12). The common factor for the first pair is a, and for the second pair is -3. This gives us: a(3a + 8) - 3(3a + 8). Now we have a common factor of (3a + 8) in both terms, so the expression can be factored as (3a + 8)(a - 3).
However, this answer seems to indicate an error since it does not match any of the options provided in the question. Checking again, we realize the product of 3 and -4 is -12 and the sum of 3 and -4 is -1, not 5. Similarly, the product of -3 and 4 is -12, and the sum is 1, which also does not match. After reassessing and recalculating the products and sums, we find that the correct factors are 4 and -3, which gives us the option: (3a - 4)(a + 3), which is option 2. Therefore, the completely factored form of 3a² + 5a - 12 is (3a - 4)(a + 3).