The expression that should be subtracted from `4x^(4)-2x^(3)-6x^(2)+x-5` so that it may be exactly divisible by `2x^(2)+x-2` is

Options

A. `3x+5`
B. `-3x-5`
C. `-3x+5`
D. `3x-5`

Answer

To find the expression that should be subtracted from \(4x^4 - 2x^3 - 6x^2 + x - 5\) so that it is exactly divisible by \(2x^2 + x - 2\), we need to perform polynomial long division and find the remainder.

### Step-by-Step Solution:

1. **Identify the Dividend and Divisor**:
- Dividend: \(4x^4 - 2x^3 - 6x^2 + x - 5\)
- Divisor: \(2x^2 + x - 2\)

2. **Perform Polynomial Long Division**:
- Divide the leading term of the dividend \(4x^4\) by the leading term of the divisor \(2x^2\):
\[ \frac{4x^4}{2x^2} = 2x^2 \]
- Multiply the entire divisor by \(2x^2\):
\[ 2x^2(2x^2 + x - 2) = 4x^4 + 2x^3 - 4x^2 \]
- Subtract this from the original polynomial:
\[ (4x^4 - 2x^3 - 6x^2 + x - 5) - (4x^4 + 2x^3 - 4x^2) = -4x^3 - 2x^2 + x - 5 \]

3. **Repeat the Division**:
- Now, take the new leading term \(-4x^3\) and divide by the leading term of the divisor \(2x^2\):
\[ \frac{-4x^3}{2x^2} = -2x \]
- Multiply the entire divisor by \(-2x\):
\[ -2x(2x^2 + x - 2) = -4x^3 - 2x^2 + 4x \]
- Subtract this from the previous result:
\[ (-4x^3 - 2x^2 + x - 5) - (-4x^3 - 2x^2 + 4x) = -3x - 5 \]

4. **Final Remainder**:
- The remainder is \(-3x - 5\).

5. **Expression to be Subtracted**:
- To make the original polynomial exactly divisible by the divisor, we need to subtract the remainder:
\[ -(-3x - 5) = 3x + 5 \]

### Conclusion:
The expression that should be subtracted from \(4x^4 - 2x^3 - 6x^2 + x - 5\) to make it exactly divisible by \(2x^2 + x - 2\) is \(3x + 5\).

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