Divide using synthetic division $$ ( x^{5}+x^{3}-x^{2} ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}4&1&0&1&-1&0&0\\& & 4& 16& 68& 268& \color{black}{1072} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{17}&\color{blue}{67}&\color{blue}{268}&\color{orangered}{1072} \end{array} $$The solution is:
$$ \frac{ x^{5}+x^{3}-x^{2} }{ x-4 } = \color{blue}{x^{4}+4x^{3}+17x^{2}+67x+268} ~+~ \frac{ \color{red}{ 1072 } }{ x-4 } $$Explanation
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&1&-1&0&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}4&\color{orangered}{ 1 }&0&1&-1&0&0\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&1&-1&0&0\\& & \color{blue}{4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}4&1&\color{orangered}{ 0 }&1&-1&0&0\\& & \color{orangered}{4} & & & & \\ \hline &1&\color{orangered}{4}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&1&-1&0&0\\& & 4& \color{blue}{16} & & & \\ \hline &1&\color{blue}{4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 16 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrrrr}4&1&0&\color{orangered}{ 1 }&-1&0&0\\& & 4& \color{orangered}{16} & & & \\ \hline &1&4&\color{orangered}{17}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 17 } = \color{blue}{ 68 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&1&-1&0&0\\& & 4& 16& \color{blue}{68} & & \\ \hline &1&4&\color{blue}{17}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 68 } = \color{orangered}{ 67 } $
$$ \begin{array}{c|rrrrrr}4&1&0&1&\color{orangered}{ -1 }&0&0\\& & 4& 16& \color{orangered}{68} & & \\ \hline &1&4&17&\color{orangered}{67}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 67 } = \color{blue}{ 268 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&1&-1&0&0\\& & 4& 16& 68& \color{blue}{268} & \\ \hline &1&4&17&\color{blue}{67}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 268 } = \color{orangered}{ 268 } $
$$ \begin{array}{c|rrrrrr}4&1&0&1&-1&\color{orangered}{ 0 }&0\\& & 4& 16& 68& \color{orangered}{268} & \\ \hline &1&4&17&67&\color{orangered}{268}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 268 } = \color{blue}{ 1072 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&1&-1&0&0\\& & 4& 16& 68& 268& \color{blue}{1072} \\ \hline &1&4&17&67&\color{blue}{268}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1072 } = \color{orangered}{ 1072 } $
$$ \begin{array}{c|rrrrrr}4&1&0&1&-1&0&\color{orangered}{ 0 }\\& & 4& 16& 68& 268& \color{orangered}{1072} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{17}&\color{blue}{67}&\color{blue}{268}&\color{orangered}{1072} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+4x^{3}+17x^{2}+67x+268 } $ with a remainder of $ \color{red}{ 1072 } $.