What is the remainder, when `( 4 x ^(4) + 10 x^(3) - 20 x ^(2) + 90)` is divide by ( x + 2) ?

Options

A. 0
B. `-6`
C. 45
D. `-90`

Answer

To find the remainder when \( 4x^4 + 10x^3 - 20x^2 + 90 \) is divided by \( x + 2 \), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). In our case, we need to find \( f(-2) \) since \( x + 2 = 0 \) gives \( x = -2 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the value of \( x \)**:
\[ f(x) = 4x^4 + 10x^3 - 20x^2 + 90 \]
We need to evaluate \( f(-2) \).

2. **Substitute \( x = -2 \) into the polynomial**:
\[ f(-2) = 4(-2)^4 + 10(-2)^3 - 20(-2)^2 + 90 \]

3. **Calculate each term**:
- Calculate \( (-2)^4 \):
\[ (-2)^4 = 16 \quad \Rightarrow \quad 4 \times 16 = 64 \]
- Calculate \( (-2)^3 \):
\[ (-2)^3 = -8 \quad \Rightarrow \quad 10 \times -8 = -80 \]
- Calculate \( (-2)^2 \):
\[ (-2)^2 = 4 \quad \Rightarrow \quad -20 \times 4 = -80 \]

4. **Combine all the terms**:
\[ f(-2) = 64 - 80 - 80 + 90 \]

5. **Perform the addition and subtraction**:
- First, combine \( 64 + 90 \):
\[ 64 + 90 = 154 \]
- Then combine \( -80 - 80 \):
\[ -80 - 80 = -160 \]
- Finally, combine \( 154 - 160 \):
\[ 154 - 160 = -6 \]

6. **Conclusion**:
The remainder when \( 4x^4 + 10x^3 - 20x^2 + 90 \) is divided by \( x + 2 \) is \( -6 \).

### Final Answer:
The remainder is \( -6 \).