Coeffieient of `x ^(2)` in `(x+3) (2-4x) (5x -6)` is

Options

A. 26
B. `-74`
C. 74
D. `-26 `

Answer

To find the coefficient of \( x^2 \) in the expression \( (x+3)(2-4x)(5x-6) \), we can follow these steps:

### Step 1: Expand the first two factors
First, we will expand \( (x + 3)(2 - 4x) \).

\[ (x + 3)(2 - 4x) = x \cdot 2 + x \cdot (-4x) + 3 \cdot 2 + 3 \cdot (-4x) \]
\[ = 2x - 4x^2 + 6 - 12x \]
\[ = -4x^2 - 10x + 6 \]

### Step 2: Multiply the result with the third factor
Next, we will multiply the result from Step 1 with \( (5x - 6) \).

\[ (-4x^2 - 10x + 6)(5x - 6) \]

We will distribute each term in \( (-4x^2 - 10x + 6) \) with \( (5x - 6) \):

1. Multiply \( -4x^2 \) with \( 5x \) and \( -6 \):
\[ -4x^2 \cdot 5x = -20x^3 \]
\[ -4x^2 \cdot (-6) = 24x^2 \]

2. Multiply \( -10x \) with \( 5x \) and \( -6 \):
\[ -10x \cdot 5x = -50x^2 \]
\[ -10x \cdot (-6) = 60x \]

3. Multiply \( 6 \) with \( 5x \) and \( -6 \):
\[ 6 \cdot 5x = 30x \]
\[ 6 \cdot (-6) = -36 \]

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

\[ -20x^3 + (24x^2 - 50x^2) + (60x + 30x) - 36 \]
\[ = -20x^3 - 26x^2 + 90x - 36 \]

### Step 4: Identify the coefficient of \( x^2 \)
From the combined expression, the coefficient of \( x^2 \) is \( -26 \).

### Final Answer
The coefficient of \( x^2 \) in the expression \( (x + 3)(2 - 4x)(5x - 6) \) is \( -26 \).

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