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

### Step-by-step Solution:

1. **Simplify \( x \)**:
\[ x = -5 + 2\sqrt{-4} = -5 + 2 \cdot 2i = -5 + 4i \]

2. **Set up the polynomial**:
We want to evaluate:
\[ P(x) = x^4 + 9x^3 + 35x^2 - x + 4 \]

3. **Find a polynomial that \( x \) satisfies**:
We can express \( x + 5 = 4i \). Squaring both sides:
\[ (x + 5)^2 = (4i)^2 \implies x^2 + 10x + 25 = -16 \implies x^2 + 10x + 41 = 0 \]

4. **Use the polynomial \( x^2 + 10x + 41 = 0 \)**:
From this, we can express \( x^2 \) in terms of \( x \):
\[ x^2 = -10x - 41 \]

5. **Calculate \( x^3 \)**:
Multiply \( x^2 \) by \( x \):
\[ x^3 = x \cdot x^2 = x(-10x - 41) = -10x^2 - 41x \]
Substitute \( x^2 \):
\[ x^3 = -10(-10x - 41) - 41x = 100x + 410 - 41x = 59x + 410 \]

6. **Calculate \( x^4 \)**:
Multiply \( x^3 \) by \( x \):
\[ x^4 = x \cdot x^3 = x(59x + 410) = 59x^2 + 410x \]
Substitute \( x^2 \):
\[ x^4 = 59(-10x - 41) + 410x = -590x - 2419 + 410x = -180x - 2419 \]

7. **Substitute \( x^4 \), \( x^3 \), and \( x^2 \) into \( P(x) \)**:
\[ P(x) = (-180x - 2419) + 9(59x + 410) + 35(-10x - 41) - x + 4 \]

Now, simplify each term:
\[ P(x) = -180x - 2419 + 531x + 3690 - 350x - 1435 - x + 4 \]

8. **Combine like terms**:
Combine the coefficients of \( x \):
\[ (-180 + 531 - 350 - 1)x + (-2419 + 3690 - 1435 + 4) \]
\[ = 0x + (-2419 + 3690 - 1435 + 4) = -2419 + 3690 - 1435 + 4 = -160 \]

Thus, the value of the expression \( x^4 + 9x^3 + 35x^2 - x + 4 \) is:
\[ \boxed{-160} \]