Divide using synthetic division $$ ( 5x^{4}-x^{3}-4x+6 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&5&-1&0&-4&6\\& & 20& 76& 304& \color{black}{1200} \\ \hline &\color{blue}{5}&\color{blue}{19}&\color{blue}{76}&\color{blue}{300}&\color{orangered}{1206} \end{array} $$The solution is:
$$ \frac{ 5x^{4}-x^{3}-4x+6 }{ x-4 } = \color{blue}{5x^{3}+19x^{2}+76x+300} ~+~ \frac{ \color{red}{ 1206 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-1&0&-4&6\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 5 }&-1&0&-4&6\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 5 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-1&0&-4&6\\& & \color{blue}{20} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 20 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}4&5&\color{orangered}{ -1 }&0&-4&6\\& & \color{orangered}{20} & & & \\ \hline &5&\color{orangered}{19}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 19 } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-1&0&-4&6\\& & 20& \color{blue}{76} & & \\ \hline &5&\color{blue}{19}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 76 } = \color{orangered}{ 76 } $
$$ \begin{array}{c|rrrrr}4&5&-1&\color{orangered}{ 0 }&-4&6\\& & 20& \color{orangered}{76} & & \\ \hline &5&19&\color{orangered}{76}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 76 } = \color{blue}{ 304 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-1&0&-4&6\\& & 20& 76& \color{blue}{304} & \\ \hline &5&19&\color{blue}{76}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 304 } = \color{orangered}{ 300 } $
$$ \begin{array}{c|rrrrr}4&5&-1&0&\color{orangered}{ -4 }&6\\& & 20& 76& \color{orangered}{304} & \\ \hline &5&19&76&\color{orangered}{300}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 300 } = \color{blue}{ 1200 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-1&0&-4&6\\& & 20& 76& 304& \color{blue}{1200} \\ \hline &5&19&76&\color{blue}{300}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 1200 } = \color{orangered}{ 1206 } $
$$ \begin{array}{c|rrrrr}4&5&-1&0&-4&\color{orangered}{ 6 }\\& & 20& 76& 304& \color{orangered}{1200} \\ \hline &\color{blue}{5}&\color{blue}{19}&\color{blue}{76}&\color{blue}{300}&\color{orangered}{1206} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{3}+19x^{2}+76x+300 } $ with a remainder of $ \color{red}{ 1206 } $.