Which is the equivalent form of `(x-3)(x^2+4x-8)` ?

Options

A. `x^3 + x^2- 20x + 24`
B. `x^3 + 7x^2 + 4x -24`
C. `x^3+ 7x^2+ 20x - 24`
D. `x^3+ x^2 + 4x + 24`

Answer

To find the equivalent form of the expression \((x-3)(x^2+4x-8)\), we will use the distributive property (also known as the FOIL method for binomials) to expand the expression step by step.

### Step-by-Step Solution:

1. **Write the expression**:
\[ (x - 3)(x^2 + 4x - 8) \]

2. **Distribute \(x\) to each term in the second polynomial**:
- Multiply \(x\) by \(x^2\):
\[ x \cdot x^2 = x^3 \]
- Multiply \(x\) by \(4x\):
\[ x \cdot 4x = 4x^2 \]
- Multiply \(x\) by \(-8\):
\[ x \cdot (-8) = -8x \]

After this step, we have:
\[ x^3 + 4x^2 - 8x \]

3. **Distribute \(-3\) to each term in the second polynomial**:
- Multiply \(-3\) by \(x^2\):
\[ -3 \cdot x^2 = -3x^2 \]
- Multiply \(-3\) by \(4x\):
\[ -3 \cdot 4x = -12x \]
- Multiply \(-3\) by \(-8\):
\[ -3 \cdot (-8) = 24 \]

After this step, we have:
\[ -3x^2 - 12x + 24 \]

4. **Combine all the terms**:
Now, we combine the results from steps 2 and 3:
\[ x^3 + 4x^2 - 8x - 3x^2 - 12x + 24 \]

5. **Combine like terms**:
- Combine \(4x^2\) and \(-3x^2\):
\[ 4x^2 - 3x^2 = 1x^2 \text{ or simply } x^2 \]
- Combine \(-8x\) and \(-12x\):
\[ -8x - 12x = -20x \]

Thus, we have:
\[ x^3 + x^2 - 20x + 24 \]

6. **Final expression**:
The equivalent form of \((x-3)(x^2+4x-8)\) is:
\[ x^3 + x^2 - 20x + 24 \]