Divide using synthetic division $$ ( 3x^{4}-16x^{3}-7x^{2}+64x-20 ) \div ( x-10 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}10&3&-16&-7&64&-20\\& & 30& 140& 1330& \color{black}{13940} \\ \hline &\color{blue}{3}&\color{blue}{14}&\color{blue}{133}&\color{blue}{1394}&\color{orangered}{13920} \end{array} $$The solution is:
$$ \dfrac{ 3x^{4}-16x^{3}-7x^{2}+64x-20 }{ x-10 } = \color{blue}{3x^{3}+14x^{2}+133x+1394} ~+~ \dfrac{ \color{red}{ 13920 } }{ x-10 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -10 = 0 $ ( $ x = \color{blue}{ 10 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&3&-16&-7&64&-20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}10&\color{orangered}{ 3 }&-16&-7&64&-20\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 3 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&3&-16&-7&64&-20\\& & \color{blue}{30} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 30 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrr}10&3&\color{orangered}{ -16 }&-7&64&-20\\& & \color{orangered}{30} & & & \\ \hline &3&\color{orangered}{14}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 14 } = \color{blue}{ 140 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&3&-16&-7&64&-20\\& & 30& \color{blue}{140} & & \\ \hline &3&\color{blue}{14}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 140 } = \color{orangered}{ 133 } $
$$ \begin{array}{c|rrrrr}10&3&-16&\color{orangered}{ -7 }&64&-20\\& & 30& \color{orangered}{140} & & \\ \hline &3&14&\color{orangered}{133}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 133 } = \color{blue}{ 1330 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&3&-16&-7&64&-20\\& & 30& 140& \color{blue}{1330} & \\ \hline &3&14&\color{blue}{133}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ 1330 } = \color{orangered}{ 1394 } $
$$ \begin{array}{c|rrrrr}10&3&-16&-7&\color{orangered}{ 64 }&-20\\& & 30& 140& \color{orangered}{1330} & \\ \hline &3&14&133&\color{orangered}{1394}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 1394 } = \color{blue}{ 13940 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&3&-16&-7&64&-20\\& & 30& 140& 1330& \color{blue}{13940} \\ \hline &3&14&133&\color{blue}{1394}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 13940 } = \color{orangered}{ 13920 } $
$$ \begin{array}{c|rrrrr}10&3&-16&-7&64&\color{orangered}{ -20 }\\& & 30& 140& 1330& \color{orangered}{13940} \\ \hline &\color{blue}{3}&\color{blue}{14}&\color{blue}{133}&\color{blue}{1394}&\color{orangered}{13920} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+14x^{2}+133x+1394 } $ with a remainder of $ \color{red}{ 13920 } $.