Coefficient of `x ^(2)` in `(x + 9) (6 - 4x) (4x -7)` is

Options

A. 216
B. `-4`
C. `-92`
D. 108

Answer

To find the coefficient of \( x^2 \) in the expression \( (x + 9)(6 - 4x)(4x - 7) \), we can follow these steps:

### Step 1: Expand the first two factors
We start by expanding \( (x + 9)(6 - 4x) \):

\[ (x + 9)(6 - 4x) = x \cdot 6 + x \cdot (-4x) + 9 \cdot 6 + 9 \cdot (-4x) \]

Calculating each term:

- \( x \cdot 6 = 6x \)
- \( x \cdot (-4x) = -4x^2 \)
- \( 9 \cdot 6 = 54 \)
- \( 9 \cdot (-4x) = -36x \)

Now combine these results:

\[ = 6x - 4x^2 + 54 - 36x \]

Combine like terms:

\[ = -4x^2 + (6x - 36x) + 54 = -4x^2 - 30x + 54 \]

### Step 2: Expand with the third factor
Now we will multiply this result by the third factor \( (4x - 7) \):

\[ (-4x^2 - 30x + 54)(4x - 7) \]

We will distribute each term in \( -4x^2 - 30x + 54 \) with \( 4x - 7 \):

1. **Multiply \( -4x^2 \) with \( 4x \) and \( -7 \)**:
- \( -4x^2 \cdot 4x = -16x^3 \)
- \( -4x^2 \cdot (-7) = 28x^2 \)

2. **Multiply \( -30x \) with \( 4x \) and \( -7 \)**:
- \( -30x \cdot 4x = -120x^2 \)
- \( -30x \cdot (-7) = 210x \)

3. **Multiply \( 54 \) with \( 4x \) and \( -7 \)**:
- \( 54 \cdot 4x = 216x \)
- \( 54 \cdot (-7) = -378 \)

### Step 3: Combine all the results
Now we combine all the terms we calculated:

- From \( -16x^3 \)
- From \( 28x^2 - 120x^2 \) gives \( 28x^2 - 120x^2 = -92x^2 \)
- From \( 210x + 216x = 426x \)
- Finally, the constant term is \( -378 \)

So, we have:

\[ -16x^3 - 92x^2 + 426x - 378 \]

### Step 4: Identify the coefficient of \( x^2 \)
The coefficient of \( x^2 \) in the expanded expression is \( -92 \).

Thus, the coefficient of \( x^2 \) in \( (x + 9)(6 - 4x)(4x - 7) \) is:

\[ \boxed{-92} \]