d) \( \left(x^{7}+2 x^{6}+x^{5}+2 x^{4}+x^{3}+2 x^{2}-x\right):\left(x^{2}+2 x\right) \)

To perform the polynomial long division of \( x^{7}+2 x^{6}+x^{5}+2 x^{4}+x^{3}+2 x^{2}-x \) by \( x^{2}+2 x \), we can follow these steps: 1. **Set up the division**: We want to divide \( x^{7}+2 x^{6}+x^{5}+2 x^{4}+x^{3}+2 x^{2}-x \) by \( x^{2}+2 x \). 2. **Divide the leading terms**: The leading term of the dividend is \( x^7 \) and the leading term of the divisor is \( x^2 \). Dividing these gives: \[ \frac{x^7}{x^2} = x^5 \] 3. **Multiply and subtract**: Multiply \( x^5 \) by the entire divisor \( x^2 + 2x \): \[ x^5(x^2 + 2x) = x^7 + 2x^6 \] Now subtract this from the original polynomial: \[ (x^{7}+2 x^{6}+x^{5}+2 x^{4}+x^{3}+2 x^{2}-x) - (x^{7}+2 x^{6}) = x^{5}+2 x^{4}+x^{3}+2 x^{2}-x \] 4. **Repeat the process**: Now we have \( x^{5}+2 x^{4}+x^{3}+2 x^{2}-x \). Divide the leading term \( x^5 \) by \( x^2 \): \[ \frac{x^5}{x^2} = x^3 \] Multiply and subtract: \[ x^3(x^2 + 2x) = x^5 + 2x^4 \] Subtract: \[ (x^{5}+2 x^{4}+x^{3}+2 x^{2}-x) - (x^{5}+2 x^{4}) = x^{3}+2 x^{2}-x \] 5. **Continue**: Now we have \( x^{3}+2 x^{2}-x \). Divide \( x^3 \) by \( x^2 \): \[ \frac{x^3}{x^2} = x \] Multiply and subtract: \[ x(x^2 + 2x) = x^3 + 2x^2 \] Subtract: \[ (x^{3}+2 x^{2}-x) - (x^{3}+2 x^{2}) = -x \] 6. **Final step**: Now we have \( -x \). Divide \( -x \) by \( x^2 \): \[ \frac{-x}{x^2} = 0 \] Since the degree of the remainder \( -x \) is less than the degree of the divisor \( x^2 + 2x \), we stop here. The result of the division is: \[ x^5 + x^3 + x + \frac{-x}{x^2 + 2x} \] Thus, the final answer is: \[ \boxed{x^5 + x^3 + x - \frac{x}{x^2 + 2x}} \]