Divide using synthetic division $$ ( -2x^{2}+x-2 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&-2&1&-2\\& & -4& \color{black}{-6} \\ \hline &\color{blue}{-2}&\color{blue}{-3}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ -2x^{2}+x-2 }{ x-2 } = \color{blue}{-2x-3} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{2}&-2&1&-2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ -2 }&1&-2\\& & & \\ \hline &\color{orangered}{-2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-2&1&-2\\& & \color{blue}{-4} & \\ \hline &\color{blue}{-2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrr}2&-2&\color{orangered}{ 1 }&-2\\& & \color{orangered}{-4} & \\ \hline &-2&\color{orangered}{-3}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-2&1&-2\\& & -4& \color{blue}{-6} \\ \hline &-2&\color{blue}{-3}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}2&-2&1&\color{orangered}{ -2 }\\& & -4& \color{orangered}{-6} \\ \hline &\color{blue}{-2}&\color{blue}{-3}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x-3 } $ with a remainder of $ \color{red}{ -8 } $.