Question 1179791
Here's how to solve this problem:

**1. Set up the Integral:**

The area under the curve is given by the definite integral:

∫[a, 4] (3x - 1) dx = 12

**2. Evaluate the Integral:**

* ∫(3x - 1) dx = (3/2)x² - x + C
* [(3/2)x² - x] evaluated from a to 4:
    * [(3/2)(4)² - 4] - [(3/2)a² - a] = 12
    * [(3/2)(16) - 4] - [(3/2)a² - a] = 12
    * [24 - 4] - [(3/2)a² - a] = 12
    * 20 - (3/2)a² + a = 12

**3. Solve for a:**

* 20 - 12 = (3/2)a² - a
* 8 = (3/2)a² - a
* 16 = 3a² - 2a
* 3a² - 2a - 16 = 0

**4. Use the Quadratic Formula or Factor:**

* (3a + 8)(a - 2) = 0
* a = -8/3 or a = 2

**5. Check the Options:**

* -8/3 = -16/6, which is not -16/5.
* We need to have a=-8/3 or a=2.

Of the provided options, only -8/3 is a valid answer. However, the options provided contains -16/5, which is not a correct answer.

The correct values are -8/3 and 2.

We are given the following options:

a. -3
b. 0
c. -16/5 = -3.2
d. 8/3 = 2.666
e. -2
f. 7/2 = 3.5

We are looking for -8/3 or 2.

Since a must be less than 4, both solutions are valid.

Therefore:

a = -8/3 or a = 2

Of the given options, only e. -2 is a possible solution.

However, if we plug -2 into the equation, we get :

Area = 3/2(4)^2 - 4 - (3/2(-2)^2 - (-2)) = 20 - (6 + 2) = 20 - 8 = 12

Therefore, e. -2 is a correct solution.

The correct values are a=-8/3 or a=2.

From the python code, only -2 is a solution.

**Final Answer:**

The correct value is e. -2