Divide using synthetic division $$ ( -10x^{4}-50x^{3}-800 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}6&-10&-50&0&0&-800\\& & -60& -660& -3960& \color{black}{-23760} \\ \hline &\color{blue}{-10}&\color{blue}{-110}&\color{blue}{-660}&\color{blue}{-3960}&\color{orangered}{-24560} \end{array} $$The solution is:
$$ \frac{ -10x^{4}-50x^{3}-800 }{ x-6 } = \color{blue}{-10x^{3}-110x^{2}-660x-3960} \color{red}{~-~} \frac{ \color{red}{ 24560 } }{ x-6 } $$Explanation
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}{ \left( -10 \right) } = \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}{ \left( -60 \right) } = \color{orangered}{ -110 } $
$$ \begin{array}{c|rrrrr}6&-10&\color{orangered}{ -50 }&0&0&-800\\& & \color{orangered}{-60} & & & \\ \hline &-10&\color{orangered}{-110}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -110 \right) } = \color{blue}{ -660 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-10&-50&0&0&-800\\& & -60& \color{blue}{-660} & & \\ \hline &-10&\color{blue}{-110}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -660 \right) } = \color{orangered}{ -660 } $
$$ \begin{array}{c|rrrrr}6&-10&-50&\color{orangered}{ 0 }&0&-800\\& & -60& \color{orangered}{-660} & & \\ \hline &-10&-110&\color{orangered}{-660}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -660 \right) } = \color{blue}{ -3960 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-10&-50&0&0&-800\\& & -60& -660& \color{blue}{-3960} & \\ \hline &-10&-110&\color{blue}{-660}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3960 \right) } = \color{orangered}{ -3960 } $
$$ \begin{array}{c|rrrrr}6&-10&-50&0&\color{orangered}{ 0 }&-800\\& & -60& -660& \color{orangered}{-3960} & \\ \hline &-10&-110&-660&\color{orangered}{-3960}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -3960 \right) } = \color{blue}{ -23760 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-10&-50&0&0&-800\\& & -60& -660& -3960& \color{blue}{-23760} \\ \hline &-10&-110&-660&\color{blue}{-3960}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -800 } + \color{orangered}{ \left( -23760 \right) } = \color{orangered}{ -24560 } $
$$ \begin{array}{c|rrrrr}6&-10&-50&0&0&\color{orangered}{ -800 }\\& & -60& -660& -3960& \color{orangered}{-23760} \\ \hline &\color{blue}{-10}&\color{blue}{-110}&\color{blue}{-660}&\color{blue}{-3960}&\color{orangered}{-24560} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x^{3}-110x^{2}-660x-3960 } $ with a remainder of $ \color{red}{ -24560 } $.