Divide using synthetic division $$ ( -x^{2}+2x-7 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&-1&2&-7\\& & -5& \color{black}{-15} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{orangered}{-22} \end{array} $$The solution is:
$$ \frac{ -x^{2}+2x-7 }{ x-5 } = \color{blue}{-x-3} \color{red}{~-~} \frac{ \color{red}{ 22 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{5}&-1&2&-7\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ -1 }&2&-7\\& & & \\ \hline &\color{orangered}{-1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-1&2&-7\\& & \color{blue}{-5} & \\ \hline &\color{blue}{-1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrr}5&-1&\color{orangered}{ 2 }&-7\\& & \color{orangered}{-5} & \\ \hline &-1&\color{orangered}{-3}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-1&2&-7\\& & -5& \color{blue}{-15} \\ \hline &-1&\color{blue}{-3}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrr}5&-1&2&\color{orangered}{ -7 }\\& & -5& \color{orangered}{-15} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{orangered}{-22} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x-3 } $ with a remainder of $ \color{red}{ -22 } $.