Divide using synthetic division $$ ( 6x^{3}-2x^{2}-2x ) \div ( x-7 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&6&-2&-2&0\\& & 42& 280& \color{black}{1946} \\ \hline &\color{blue}{6}&\color{blue}{40}&\color{blue}{278}&\color{orangered}{1946} \end{array} $$The solution is:
$$ \dfrac{ 6x^{3}-2x^{2}-2x }{ x-7 } = \color{blue}{6x^{2}+40x+278} ~+~ \dfrac{ \color{red}{ 1946 } }{ x-7 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{7}&6&-2&-2&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 6 }&-2&-2&0\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 6 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&6&-2&-2&0\\& & \color{blue}{42} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 42 } = \color{orangered}{ 40 } $
$$ \begin{array}{c|rrrr}7&6&\color{orangered}{ -2 }&-2&0\\& & \color{orangered}{42} & & \\ \hline &6&\color{orangered}{40}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 40 } = \color{blue}{ 280 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&6&-2&-2&0\\& & 42& \color{blue}{280} & \\ \hline &6&\color{blue}{40}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 280 } = \color{orangered}{ 278 } $
$$ \begin{array}{c|rrrr}7&6&-2&\color{orangered}{ -2 }&0\\& & 42& \color{orangered}{280} & \\ \hline &6&40&\color{orangered}{278}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 278 } = \color{blue}{ 1946 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&6&-2&-2&0\\& & 42& 280& \color{blue}{1946} \\ \hline &6&40&\color{blue}{278}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1946 } = \color{orangered}{ 1946 } $
$$ \begin{array}{c|rrrr}7&6&-2&-2&\color{orangered}{ 0 }\\& & 42& 280& \color{orangered}{1946} \\ \hline &\color{blue}{6}&\color{blue}{40}&\color{blue}{278}&\color{orangered}{1946} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+40x+278 } $ with a remainder of $ \color{red}{ 1946 } $.