(2x^3-x^2-x+1):(x^2+2x)

To divide the polynomial \(2x^3 - x^2 - x + 1\) by \(x^2 + 2x\), we can use polynomial long division. Here are the steps:

1. **Setup the Division**:
\[
\frac{2x^3 - x^2 - x + 1}{x^2 + 2x}
\]

2. **Divide the Leading Terms**:
Divide the leading term of the numerator \(2x^3\) by the leading term of the denominator \(x^2\):
\[
\frac{2x^3}{x^2} = 2x
\]

3. **Multiply and Subtract**:
Multiply \(2x\) by the entire divisor \(x^2 + 2x\):
\[
2x \cdot (x^2 + 2x) = 2x^3 + 4x^2
\]
Subtract this result from the original polynomial:
\[
(2x^3 - x^2 - x + 1) - (2x^3 + 4x^2) = -5x^2 - x + 1
\]

4. **Repeat the Process**:
Now, divide the new leading term \(-5x^2\) by the leading term of the divisor \(x^2\):
\[
\frac{-5x^2}{x^2} = -5
\]
Multiply \(-5\) by the entire divisor \(x^2 + 2x\):
\[
-5 \cdot (x^2 + 2x) = -5x^2 - 10x
\]
Subtract this result from the current polynomial:
\[
(-5x^2 - x + 1) - (-5x^2 - 10x) = 9x + 1
\]

5. **Write the Final Result**:
The quotient is \(2x - 5\) and the remainder is \(9x + 1\). So, the division can be expressed as:
\[
\frac{2x^3 - x^2 - x + 1}{x^2 + 2x} = 2x - 5 + \frac{9x + 1}{x^2 + 2x}
\]

Therefore, the result of the division is:
\[
2x - 5 + \frac{9x + 1}{x^2 + 2x}
\]