Divide using synthetic division $$ ( -x^{2}-5x+6 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&-1&-5&6\\& & -2& \color{black}{-14} \\ \hline &\color{blue}{-1}&\color{blue}{-7}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ -x^{2}-5x+6 }{ x-2 } = \color{blue}{-x-7} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x-2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{2}&-1&-5&6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ -1 }&-5&6\\& & & \\ \hline &\color{orangered}{-1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-1&-5&6\\& & \color{blue}{-2} & \\ \hline &\color{blue}{-1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrr}2&-1&\color{orangered}{ -5 }&6\\& & \color{orangered}{-2} & \\ \hline &-1&\color{orangered}{-7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-1&-5&6\\& & -2& \color{blue}{-14} \\ \hline &-1&\color{blue}{-7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}2&-1&-5&\color{orangered}{ 6 }\\& & -2& \color{orangered}{-14} \\ \hline &\color{blue}{-1}&\color{blue}{-7}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x-7 } $ with a remainder of $ \color{red}{ -8 } $.