Evaluate `x^(4) -4x^(3)+4x^(2)+8x+44`, when `x=3+2i`

Answer

To evaluate the expression \( x^4 - 4x^3 + 4x^2 + 8x + 44 \) when \( x = 3 + 2i \), we will follow these steps:

### Step 1: Substitute \( x \)
We start by substituting \( x = 3 + 2i \) into the expression.

\[ x^4 - 4x^3 + 4x^2 + 8x + 44 \]

### Step 2: Calculate \( x^2 \)
First, we calculate \( x^2 \):

\[ x^2 = (3 + 2i)^2 = 3^2 + 2 \cdot 3 \cdot 2i + (2i)^2 = 9 + 12i - 4 = 5 + 12i \]

### Step 3: Calculate \( x^3 \)
Next, we calculate \( x^3 \):

\[ x^3 = (3 + 2i)(5 + 12i) = 3 \cdot 5 + 3 \cdot 12i + 2i \cdot 5 + 2i \cdot 12i \]
\[ = 15 + 36i + 10i - 24 = -9 + 46i \]

### Step 4: Calculate \( x^4 \)
Now, we calculate \( x^4 \):

\[ x^4 = (3 + 2i)(-9 + 46i) = 3 \cdot -9 + 3 \cdot 46i + 2i \cdot -9 + 2i \cdot 46i \]
\[ = -27 + 138i - 18i - 92 = -119 + 120i \]

### Step 5: Substitute \( x^2, x^3, x^4 \) into the expression
Now we substitute \( x^2, x^3, x^4 \) back into the original expression:

\[ (-119 + 120i) - 4(-9 + 46i) + 4(5 + 12i) + 8(3 + 2i) + 44 \]

### Step 6: Simplify each term
Calculating each term:

1. \( x^4 = -119 + 120i \)
2. \( -4x^3 = -4(-9 + 46i) = 36 - 184i \)
3. \( 4x^2 = 4(5 + 12i) = 20 + 48i \)
4. \( 8x = 8(3 + 2i) = 24 + 16i \)
5. Constant term = 44

### Step 7: Combine all terms
Now we combine all the terms:

\[ (-119 + 120i) + (36 - 184i) + (20 + 48i) + (24 + 16i) + 44 \]

Combining the real parts:

\[ -119 + 36 + 20 + 24 + 44 = 5 \]

Combining the imaginary parts:

\[ 120i - 184i + 48i + 16i = 0i \]

### Final Result
Thus, the value of the expression when \( x = 3 + 2i \) is:

\[ \boxed{5} \]