Find the value of `x^4+9x^3+35 x^2-x+4` for `x=-5+2sqrt(-4).`

Explanation

To find the value of \( x^4 + 9x^3 + 35x^2 - x + 4 \) for \( x = -5 + 2\sqrt{-4} \), we will follow these steps:

### Step 1: Simplify \( x \)
Given:
\[ x = -5 + 2\sqrt{-4} \]
We can simplify \( \sqrt{-4} \):
\[ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \]
Thus, we have:
\[ x = -5 + 2(2i) = -5 + 4i \]

### Step 2: Find \( x + 5 \)
Now, we will find \( x + 5 \):
\[ x + 5 = (-5 + 4i) + 5 = 4i \]

### Step 3: Square \( x + 5 \)
Next, we will square \( x + 5 \):
\[ (x + 5)^2 = (4i)^2 = 16(-1) = -16 \]

### Step 4: Form the quadratic equation
From \( (x + 5)^2 = -16 \), we can write:
\[ x^2 + 10x + 25 = -16 \]
This simplifies to:
\[ x^2 + 10x + 41 = 0 \]

### Step 5: Substitute into the polynomial
Now we substitute \( x^2 + 10x + 41 = 0 \) into the polynomial \( x^4 + 9x^3 + 35x^2 - x + 4 \).

We can express \( x^4 \) and \( x^3 \) in terms of \( x^2 \) and \( x \):
1. From \( x^2 + 10x + 41 = 0 \), we have \( x^2 = -10x - 41 \).
2. To find \( x^3 \), we multiply \( x^2 \) by \( x \):
\[ x^3 = x \cdot x^2 = x(-10x - 41) = -10x^2 - 41x \]
Substitute \( x^2 \):
\[ x^3 = -10(-10x - 41) - 41x = 100x + 410 - 41x = 59x + 410 \]
3. To find \( x^4 \), we multiply \( x^3 \) by \( x \):
\[ x^4 = x \cdot x^3 = x(59x + 410) = 59x^2 + 410x \]
Substitute \( x^2 \):
\[ x^4 = 59(-10x - 41) + 410x = -590x - 2419 + 410x = -180x - 2419 \]

### Step 6: Substitute \( x^4 \) and \( x^3 \) into the polynomial
Now we substitute \( x^4 \) and \( x^3 \) into the polynomial:
\[ x^4 + 9x^3 + 35x^2 - x + 4 = (-180x - 2419) + 9(59x + 410) + 35(-10x - 41) - x + 4 \]

### Step 7: Simplify the expression
Now we simplify:
\[ = -180x - 2419 + 531x + 3690 - 350x - 1435 - x + 4 \]
Combine like terms:
\[ (-180x + 531x - 350x - x) + (-2419 + 3690 - 1435 + 4) \]
\[ = 0x + (-2419 + 3690 - 1435 + 4) = -160 \]

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