Divide using synthetic division $$ ( -x^{2}+9x-14 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&-1&9&-14\\& & -2& \color{black}{14} \\ \hline &\color{blue}{-1}&\color{blue}{7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{2}+9x-14 }{ x-2 } = \color{blue}{-x+7} $$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.
$$ \begin{array}{c|rrr}\color{blue}{2}&-1&9&-14\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ -1 }&9&-14\\& & & \\ \hline &\color{orangered}{-1}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-1&9&-14\\& & \color{blue}{-2} & \\ \hline &\color{blue}{-1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrr}2&-1&\color{orangered}{ 9 }&-14\\& & \color{orangered}{-2} & \\ \hline &-1&\color{orangered}{7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 7 } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-1&9&-14\\& & -2& \color{blue}{14} \\ \hline &-1&\color{blue}{7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 14 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}2&-1&9&\color{orangered}{ -14 }\\& & -2& \color{orangered}{14} \\ \hline &\color{blue}{-1}&\color{blue}{7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x+7 } $ with a remainder of $ \color{red}{ 0 } $.