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