What is the remainder when `3x^(4)-2x^(3)-20x^(2)-12` is divided by x+2?

Options

A. -60
B. -36
C. -28
D. -6

Explanation

To find the remainder when the polynomial \(3x^4 - 2x^3 - 20x^2 - 12\) is divided by \(x + 2\), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of a polynomial \(f(x)\) when divided by \(x - c\) is equal to \(f(c)\). In this case, we will evaluate the polynomial at \(c = -2\).

### Step-by-step Solution:

1. **Identify the polynomial**:
\[ f(x) = 3x^4 - 2x^3 - 20x^2 - 12 \]

2. **Substitute \(x = -2\)** into the polynomial:
\[ f(-2) = 3(-2)^4 - 2(-2)^3 - 20(-2)^2 - 12 \]

3. **Calculate each term**:
- Calculate \(3(-2)^4\):
\[ 3 \times 16 = 48 \]
- Calculate \(-2(-2)^3\):
\[ -2 \times -8 = 16 \]
- Calculate \(-20(-2)^2\):
\[ -20 \times 4 = -80 \]
- The constant term is \(-12\).

4. **Combine all the results**:
\[ f(-2) = 48 + 16 - 80 - 12 \]

5. **Perform the addition and subtraction**:
- First, combine \(48 + 16\):
\[ 48 + 16 = 64 \]
- Then, subtract \(80\):
\[ 64 - 80 = -16 \]
- Finally, subtract \(12\):
\[ -16 - 12 = -28 \]

6. **Conclusion**:
The remainder when \(3x^4 - 2x^3 - 20x^2 - 12\) is divided by \(x + 2\) is \(-28\).

### Final Answer:
The remainder is \(-28\).

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