If `f(x) =x^4 + 3x ^2 - 6x -2 ` then the coefficient of ` x^3 ` in ` f(x +1)` is

Options

A. 24
B. `-24`
C. `4`
D. `-4`

Answer

To find the coefficient of \( x^3 \) in \( f(x + 1) \) where \( f(x) = x^4 + 3x^2 - 6x - 2 \), we will follow these steps:

### Step 1: Substitute \( x + 1 \) into the function \( f(x) \)

We start by substituting \( x + 1 \) into the function \( f(x) \):

\[ f(x + 1) = (x + 1)^4 + 3(x + 1)^2 - 6(x + 1) - 2 \]

### Step 2: Expand each term

Now we will expand each term separately.

1. **Expand \( (x + 1)^4 \)** using the binomial theorem:

\[ (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1 \]

2. **Expand \( 3(x + 1)^2 \)**:

\[ 3(x + 1)^2 = 3(x^2 + 2x + 1) = 3x^2 + 6x + 3 \]

3. **Expand \( -6(x + 1) \)**:

\[ -6(x + 1) = -6x - 6 \]

4. **The constant term remains as it is**:

\[ -2 \]

### Step 3: Combine all the expanded terms

Now we combine all the expanded terms:

\[ f(x + 1) = (x^4 + 4x^3 + 6x^2 + 4x + 1) + (3x^2 + 6x + 3) - (6x + 6) - 2 \]

Combining like terms:

- **For \( x^4 \)**: \( x^4 \)
- **For \( x^3 \)**: \( 4x^3 \)
- **For \( x^2 \)**: \( 6x^2 + 3x^2 = 9x^2 \)
- **For \( x \)**: \( 4x + 6x - 6x = 4x \)
- **Constant terms**: \( 1 + 3 - 6 - 2 = -4 \)

So, we have:

\[ f(x + 1) = x^4 + 4x^3 + 9x^2 + 4x - 4 \]

### Step 4: Identify the coefficient of \( x^3 \)

From the final expression, we can see that the coefficient of \( x^3 \) is \( 4 \).

### Final Answer

The coefficient of \( x^3 \) in \( f(x + 1) \) is \( \boxed{4} \).

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