Coefficient of `x ^(2)` in `(x +5) (3-2x)(4x -2)` is

Options

A. A)24
B. B)32
C. C)`-24`
D. D)`-32`

Answer

To find the coefficient of \( x^2 \) in the expression \( (x + 5)(3 - 2x)(4x - 2) \), we can follow these steps:

### Step 1: Expand the first two factors
We start by multiplying the first two factors:
\[ (x + 5)(3 - 2x) \]
Using the distributive property (FOIL method):
\[ = x \cdot 3 + x \cdot (-2x) + 5 \cdot 3 + 5 \cdot (-2x) \]
\[ = 3x - 2x^2 + 15 - 10x \]
Combining like terms:
\[ = -2x^2 - 7x + 15 \]

### Step 2: Multiply the result with the third factor
Now we multiply the result by the third factor \( (4x - 2) \):
\[ (-2x^2 - 7x + 15)(4x - 2) \]
We will distribute each term in the first polynomial:
1. Multiply \( -2x^2 \) by \( 4x \) and \( -2 \):
\[ -2x^2 \cdot 4x = -8x^3 \]
\[ -2x^2 \cdot (-2) = 4x^2 \]
2. Multiply \( -7x \) by \( 4x \) and \( -2 \):
\[ -7x \cdot 4x = -28x^2 \]
\[ -7x \cdot (-2) = 14x \]
3. Multiply \( 15 \) by \( 4x \) and \( -2 \):
\[ 15 \cdot 4x = 60x \]
\[ 15 \cdot (-2) = -30 \]

### Step 3: Combine all the terms
Now we combine all the terms:
\[ -8x^3 + (4x^2 - 28x^2) + (14x + 60x) - 30 \]
This simplifies to:
\[ -8x^3 - 24x^2 + 74x - 30 \]

### Step 4: Identify the coefficient of \( x^2 \)
From the combined expression, the coefficient of \( x^2 \) is:
\[ -24 \]

### Final Answer
Thus, the coefficient of \( x^2 \) in the expression \( (x + 5)(3 - 2x)(4x - 2) \) is:
\[ \boxed{-24} \]