Multiply :
`-8x` and `4 - 2x - x^(2)`

Answer

To solve the problem of multiplying `-8x` and `4 - 2x - x^2`, we will follow these steps:

### Step-by-Step Solution:

1. **Write the expression clearly**:
We need to multiply `-8x` with the expression `(4 - 2x - x^2)`.
This can be written as:
\[ -8x \times (4 - 2x - x^2) \]

2. **Distribute `-8x` to each term in the parentheses**:
We will multiply `-8x` by each term inside the parentheses one by one.

3. **Multiply `-8x` by `4`**:
\[ -8x \times 4 = -32x \]

4. **Multiply `-8x` by `-2x`**:
\[ -8x \times -2x = 16x^2 \quad (\text{since the product of two negatives is positive}) \]

5. **Multiply `-8x` by `-x^2`**:
\[ -8x \times -x^2 = 8x^3 \quad (\text{again, the product of two negatives is positive}) \]

6. **Combine all the results**:
Now we combine all the products we calculated:
\[ -32x + 16x^2 + 8x^3 \]

7. **Rearrange the terms in standard form**:
The standard form of a polynomial is usually written in descending order of the powers of `x`. Thus, we write:
\[ 8x^3 + 16x^2 - 32x \]

### Final Answer:
\[ 8x^3 + 16x^2 - 32x \]