If `x= -5 + sqrt(-16)`, find the value of `x^(4) + 9x^(3) + 35x^(2)-x + 4`

Answer

To solve the problem, we need to find the value of the expression \( x^4 + 9x^3 + 35x^2 - x + 4 \) given that \( x = -5 + \sqrt{-16} \).

### Step-by-Step Solution:

1. **Simplify \( x \)**:
\[ x = -5 + \sqrt{-16} = -5 + 4i \]
Here, we used the fact that \( \sqrt{-16} = 4i \).

2. **Formulate the equation**:
We can rearrange the expression for \( x \):
\[ x + 5 = 4i \]
Squaring both sides gives:
\[ (x + 5)^2 = (4i)^2 \]
Expanding both sides:
\[ x^2 + 10x + 25 = -16 \]
Rearranging this leads to:
\[ x^2 + 10x + 41 = 0 \]

3. **Use polynomial long division**:
We will divide the polynomial \( x^4 + 9x^3 + 35x^2 - x + 4 \) by \( x^2 + 10x + 41 \).

- **First Division**:
- Divide \( x^4 \) by \( x^2 \) to get \( x^2 \).
- Multiply \( x^2 \) by \( x^2 + 10x + 41 \):
\[ x^4 + 10x^3 + 41x^2 \]
- Subtract from the original polynomial:
\[ (x^4 + 9x^3 + 35x^2 - x + 4) - (x^4 + 10x^3 + 41x^2) = -x^3 - 6x^2 - x + 4 \]

- **Second Division**:
- Divide \( -x^3 \) by \( x^2 \) to get \( -x \).
- Multiply \( -x \) by \( x^2 + 10x + 41 \):
\[ -x^3 - 10x^2 - 41x \]
- Subtract:
\[ (-x^3 - 6x^2 - x + 4) - (-x^3 - 10x^2 - 41x) = 4x^2 + 40x + 4 \]

- **Third Division**:
- Divide \( 4x^2 \) by \( x^2 \) to get \( 4 \).
- Multiply \( 4 \) by \( x^2 + 10x + 41 \):
\[ 4x^2 + 40x + 164 \]
- Subtract:
\[ (4x^2 + 40x + 4) - (4x^2 + 40x + 164) = -160 \]

4. **Final Result**:
The polynomial can be expressed as:
\[ x^4 + 9x^3 + 35x^2 - x + 4 = (x^2 + 10x + 41)(x^2 - x + 4) - 160 \]
Since \( x^2 + 10x + 41 = 0 \) (from our earlier step), we have:
\[ (0)(x^2 - x + 4) - 160 = -160 \]

Thus, the value of \( x^4 + 9x^3 + 35x^2 - x + 4 \) is **\(-160\)**.