`Z=-5+4i `then `Z^4 +9Z^3 +35Z^2 – Z + 4 =`

Explanation

To solve the equation \( Z^4 + 9Z^3 + 35Z^2 - Z + 4 \) where \( Z = -5 + 4i \), we will follow these steps:

### Step 1: Calculate \( Z^2 \)
Given \( Z = -5 + 4i \), we first calculate \( Z^2 \):
\[ Z^2 = (-5 + 4i)^2 = (-5)^2 + 2(-5)(4i) + (4i)^2 \]
\[ = 25 - 40i + 16i^2 \]
Since \( i^2 = -1 \), we have:
\[ = 25 - 40i - 16 = 9 - 40i \]

### Step 2: Calculate \( Z^3 \)
Next, we calculate \( Z^3 \) using \( Z^2 \):
\[ Z^3 = Z \cdot Z^2 = (-5 + 4i)(9 - 40i) \]
Using the distributive property:
\[ = -5 \cdot 9 + (-5)(-40i) + 4i \cdot 9 + 4i \cdot (-40i) \]
\[ = -45 + 200i + 36i - 160i^2 \]
Again, substituting \( i^2 = -1 \):
\[ = -45 + 236i + 160 = 115 + 236i \]

### Step 3: Calculate \( Z^4 \)
Now, we calculate \( Z^4 \):
\[ Z^4 = Z \cdot Z^3 = (-5 + 4i)(115 + 236i) \]
Using the distributive property:
\[ = -5 \cdot 115 + (-5)(236i) + 4i \cdot 115 + 4i \cdot 236i \]
\[ = -575 - 1180i + 460i + 944i^2 \]
Substituting \( i^2 = -1 \):
\[ = -575 - 1180i + 460i - 944 = -1519 - 720i \]

### Step 4: Substitute values into the polynomial
Now we substitute \( Z^4, Z^3, Z^2, Z \) into the original polynomial:
\[ Z^4 + 9Z^3 + 35Z^2 - Z + 4 \]
Substituting the values we found:
\[ = (-1519 - 720i) + 9(115 + 236i) + 35(9 - 40i) - (-5 + 4i) + 4 \]

Calculating each term:
\[ = (-1519 - 720i) + (1035 + 2124i) + (315 - 1400i) + (5 - 4i) + 4 \]
Combining real parts:
\[ = -1519 + 1035 + 315 + 5 + 4 = -160 \]
Combining imaginary parts:
\[ = -720i + 2124i - 1400i - 4i = 0i \]

### Final Result
Thus, the final result is:
\[ Z^4 + 9Z^3 + 35Z^2 - Z + 4 = -160 \]