`(2x ^(2) + 3x -4) (3x +2) = 6x ^(3) + ax ^(2) -6x-8`
In the above equation, a is a constant. If the equation is true for all value of x, what is the value of a ?

Options

A. 4
B. 9
C. 13
D. 16

Answer

To solve the equation \((2x^2 + 3x - 4)(3x + 2) = 6x^3 + ax^2 - 6x - 8\) and find the value of \(a\), we will follow these steps:

### Step 1: Expand the left-hand side of the equation
We start by expanding the left-hand side of the equation:

\[ (2x^2 + 3x - 4)(3x + 2) \]

Using the distributive property (also known as the FOIL method for binomials), we calculate:

1. \(2x^2 \cdot 3x = 6x^3\)
2. \(2x^2 \cdot 2 = 4x^2\)
3. \(3x \cdot 3x = 9x^2\)
4. \(3x \cdot 2 = 6x\)
5. \(-4 \cdot 3x = -12x\)
6. \(-4 \cdot 2 = -8\)

Now, we combine these results:

\[ 6x^3 + (4x^2 + 9x^2) + (6x - 12x) - 8 \]

### Step 2: Combine like terms
Now we combine the like terms:

- The \(x^2\) terms: \(4x^2 + 9x^2 = 13x^2\)
- The \(x\) terms: \(6x - 12x = -6x\)

So, the left-hand side simplifies to:

\[ 6x^3 + 13x^2 - 6x - 8 \]

### Step 3: Set the left-hand side equal to the right-hand side
Now we have:

\[ 6x^3 + 13x^2 - 6x - 8 = 6x^3 + ax^2 - 6x - 8 \]

### Step 4: Compare coefficients
Since the equation must hold for all values of \(x\), we can compare the coefficients of the corresponding terms on both sides:

- Coefficient of \(x^3\): \(6 = 6\) (This is true)
- Coefficient of \(x^2\): \(13 = a\)
- Coefficient of \(x\): \(-6 = -6\) (This is true)
- Constant term: \(-8 = -8\) (This is true)

### Step 5: Solve for \(a\)
From the comparison of the coefficients of \(x^2\):

\[ a = 13 \]

Thus, the value of \(a\) is \(13\).

### Final Answer
The value of \(a\) is \(13\).

---