Answer
quotient $x^6-2x^5+5x^4-10x^3+10x^2-20x+40$ and remainder $r(x)=\frac{-68}{x+2}$
Work Step by Step
Step 1. The coefficients of the dividend can be identified in order as $\{1,0,1,0,1,0,0,12\}$ and the divisor is $x+2$; use synthetic division as shown in the figure to get the quotient and the remainder.
Step 2. We can identify the result as $\frac{x^7+x^5-10x^3+12}{x+2}=x^6-2x^5+5x^4-10x^3+10x^2-20x+40+\frac{-68}{x+2}$ with the quotient as $x^6-2x^5+5x^4-10x^3+10x^2-20x+40$ and the remainder as $r(x)=\frac{-68}{x+2}$
