If `f(x) = x^3 + 3x^2 - 4x` then `f(x + 1)` is :

Options

A. `x^3 + 7x^2 - 5x`
B. `x^3 + 6x^2 + 5x`
C. `(x + 1)^3`
D. none of these

Answer

To find \( f(x + 1) \) for the function \( f(x) = x^3 + 3x^2 - 4x \), we will substitute \( x + 1 \) into the function and simplify it step by step.

### Step-by-Step Solution:

1. **Substituting \( x + 1 \) into \( f(x) \)**:
\[ f(x + 1) = (x + 1)^3 + 3(x + 1)^2 - 4(x + 1) \]

2. **Expanding \( (x + 1)^3 \)**:
Using the binomial expansion formula:
\[ (x + 1)^3 = x^3 + 3x^2 + 3x + 1 \]

3. **Expanding \( 3(x + 1)^2 \)**:
First, calculate \( (x + 1)^2 \):
\[ (x + 1)^2 = x^2 + 2x + 1 \]
Now multiply by 3:
\[ 3(x + 1)^2 = 3(x^2 + 2x + 1) = 3x^2 + 6x + 3 \]

4. **Expanding \( -4(x + 1) \)**:
\[ -4(x + 1) = -4x - 4 \]

5. **Combining all the expanded terms**:
Now, we combine all the expanded expressions:
\[ f(x + 1) = (x^3 + 3x^2 + 3x + 1) + (3x^2 + 6x + 3) + (-4x - 4) \]

6. **Combining like terms**:
- For \( x^3 \): \( x^3 \)
- For \( x^2 \): \( 3x^2 + 3x^2 = 6x^2 \)
- For \( x \): \( 3x + 6x - 4x = 5x \)
- For constant terms: \( 1 + 3 - 4 = 0 \)

Therefore, we have:
\[ f(x + 1) = x^3 + 6x^2 + 5x \]

### Final Result:
\[ f(x + 1) = x^3 + 6x^2 + 5x \]