Divide using synthetic division $$ ( -x^{6}+x^{4}-x^{3}-x+2 ) \div ( x-5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}5&-1&0&1&-1&0&-1&2\\& & -5& -25& -120& -605& -3025& \color{black}{-15130} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-24}&\color{blue}{-121}&\color{blue}{-605}&\color{blue}{-3026}&\color{orangered}{-15128} \end{array} $$The solution is:
$$ \dfrac{ -x^{6}+x^{4}-x^{3}-x+2 }{ x-5 } = \color{blue}{-x^{5}-5x^{4}-24x^{3}-121x^{2}-605x-3026} \color{red}{~-~} \dfrac{ \color{red}{ 15128 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{5}&-1&0&1&-1&0&-1&2\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}5&\color{orangered}{ -1 }&0&1&-1&0&-1&2\\& & & & & & & \\ \hline &\color{orangered}{-1}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{5}&-1&0&1&-1&0&-1&2\\& & \color{blue}{-5} & & & & & \\ \hline &\color{blue}{-1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrrr}5&-1&\color{orangered}{ 0 }&1&-1&0&-1&2\\& & \color{orangered}{-5} & & & & & \\ \hline &-1&\color{orangered}{-5}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{5}&-1&0&1&-1&0&-1&2\\& & -5& \color{blue}{-25} & & & & \\ \hline &-1&\color{blue}{-5}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrrrr}5&-1&0&\color{orangered}{ 1 }&-1&0&-1&2\\& & -5& \color{orangered}{-25} & & & & \\ \hline &-1&-5&\color{orangered}{-24}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ -120 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{5}&-1&0&1&-1&0&-1&2\\& & -5& -25& \color{blue}{-120} & & & \\ \hline &-1&-5&\color{blue}{-24}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -120 \right) } = \color{orangered}{ -121 } $
$$ \begin{array}{c|rrrrrrr}5&-1&0&1&\color{orangered}{ -1 }&0&-1&2\\& & -5& -25& \color{orangered}{-120} & & & \\ \hline &-1&-5&-24&\color{orangered}{-121}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -121 \right) } = \color{blue}{ -605 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{5}&-1&0&1&-1&0&-1&2\\& & -5& -25& -120& \color{blue}{-605} & & \\ \hline &-1&-5&-24&\color{blue}{-121}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -605 \right) } = \color{orangered}{ -605 } $
$$ \begin{array}{c|rrrrrrr}5&-1&0&1&-1&\color{orangered}{ 0 }&-1&2\\& & -5& -25& -120& \color{orangered}{-605} & & \\ \hline &-1&-5&-24&-121&\color{orangered}{-605}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -605 \right) } = \color{blue}{ -3025 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{5}&-1&0&1&-1&0&-1&2\\& & -5& -25& -120& -605& \color{blue}{-3025} & \\ \hline &-1&-5&-24&-121&\color{blue}{-605}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -3025 \right) } = \color{orangered}{ -3026 } $
$$ \begin{array}{c|rrrrrrr}5&-1&0&1&-1&0&\color{orangered}{ -1 }&2\\& & -5& -25& -120& -605& \color{orangered}{-3025} & \\ \hline &-1&-5&-24&-121&-605&\color{orangered}{-3026}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -3026 \right) } = \color{blue}{ -15130 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{5}&-1&0&1&-1&0&-1&2\\& & -5& -25& -120& -605& -3025& \color{blue}{-15130} \\ \hline &-1&-5&-24&-121&-605&\color{blue}{-3026}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -15130 \right) } = \color{orangered}{ -15128 } $
$$ \begin{array}{c|rrrrrrr}5&-1&0&1&-1&0&-1&\color{orangered}{ 2 }\\& & -5& -25& -120& -605& -3025& \color{orangered}{-15130} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-24}&\color{blue}{-121}&\color{blue}{-605}&\color{blue}{-3026}&\color{orangered}{-15128} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{5}-5x^{4}-24x^{3}-121x^{2}-605x-3026 } $ with a remainder of $ \color{red}{ -15128 } $.