Divide using synthetic division $$ ( x^{3}+5x^{2}-32x-7 ) \div ( x-4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&1&5&-32&-7\\& & 4& 36& \color{black}{16} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{4}&\color{orangered}{9} \end{array} $$The solution is:
$$ \dfrac{ x^{3}+5x^{2}-32x-7 }{ x-4 } = \color{blue}{x^{2}+9x+4} ~+~ \dfrac{ \color{red}{ 9 } }{ x-4 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&5&-32&-7\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 1 }&5&-32&-7\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&5&-32&-7\\& & \color{blue}{4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 4 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}4&1&\color{orangered}{ 5 }&-32&-7\\& & \color{orangered}{4} & & \\ \hline &1&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 9 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&5&-32&-7\\& & 4& \color{blue}{36} & \\ \hline &1&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 36 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}4&1&5&\color{orangered}{ -32 }&-7\\& & 4& \color{orangered}{36} & \\ \hline &1&9&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&1&5&-32&-7\\& & 4& 36& \color{blue}{16} \\ \hline &1&9&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 16 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}4&1&5&-32&\color{orangered}{ -7 }\\& & 4& 36& \color{orangered}{16} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{4}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+9x+4 } $ with a remainder of $ \color{red}{ 9 } $.