Divide using synthetic division $$ ( 10x^{4}-50x^{3}-800 ) \div ( x-6 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}6&10&-50&0&0&-800\\& & 60& 60& 360& \color{black}{2160} \\ \hline &\color{blue}{10}&\color{blue}{10}&\color{blue}{60}&\color{blue}{360}&\color{orangered}{1360} \end{array} $$The solution is:
$$ \dfrac{ 10x^{4}-50x^{3}-800 }{ x-6 } = \color{blue}{10x^{3}+10x^{2}+60x+360} ~+~ \dfrac{ \color{red}{ 1360 } }{ x-6 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&10&-50&0&0&-800\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}6&\color{orangered}{ 10 }&-50&0&0&-800\\& & & & & \\ \hline &\color{orangered}{10}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 10 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&10&-50&0&0&-800\\& & \color{blue}{60} & & & \\ \hline &\color{blue}{10}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -50 } + \color{orangered}{ 60 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrr}6&10&\color{orangered}{ -50 }&0&0&-800\\& & \color{orangered}{60} & & & \\ \hline &10&\color{orangered}{10}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 10 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&10&-50&0&0&-800\\& & 60& \color{blue}{60} & & \\ \hline &10&\color{blue}{10}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 60 } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrrrr}6&10&-50&\color{orangered}{ 0 }&0&-800\\& & 60& \color{orangered}{60} & & \\ \hline &10&10&\color{orangered}{60}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 60 } = \color{blue}{ 360 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&10&-50&0&0&-800\\& & 60& 60& \color{blue}{360} & \\ \hline &10&10&\color{blue}{60}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 360 } = \color{orangered}{ 360 } $
$$ \begin{array}{c|rrrrr}6&10&-50&0&\color{orangered}{ 0 }&-800\\& & 60& 60& \color{orangered}{360} & \\ \hline &10&10&60&\color{orangered}{360}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 360 } = \color{blue}{ 2160 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&10&-50&0&0&-800\\& & 60& 60& 360& \color{blue}{2160} \\ \hline &10&10&60&\color{blue}{360}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -800 } + \color{orangered}{ 2160 } = \color{orangered}{ 1360 } $
$$ \begin{array}{c|rrrrr}6&10&-50&0&0&\color{orangered}{ -800 }\\& & 60& 60& 360& \color{orangered}{2160} \\ \hline &\color{blue}{10}&\color{blue}{10}&\color{blue}{60}&\color{blue}{360}&\color{orangered}{1360} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x^{3}+10x^{2}+60x+360 } $ with a remainder of $ \color{red}{ 1360 } $.