Divide using synthetic division $$ ( 5x^{2}-10x-47 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&5&-10&-47\\& & 20& \color{black}{40} \\ \hline &\color{blue}{5}&\color{blue}{10}&\color{orangered}{-7} \end{array} $$The solution is:
$$ \frac{ 5x^{2}-10x-47 }{ x-4 } = \color{blue}{5x+10} \color{red}{~-~} \frac{ \color{red}{ 7 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&5&-10&-47\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 5 }&-10&-47\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&5&-10&-47\\& & \color{blue}{20} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 20 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrr}4&5&\color{orangered}{ -10 }&-47\\& & \color{orangered}{20} & \\ \hline &5&\color{orangered}{10}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 10 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&5&-10&-47\\& & 20& \color{blue}{40} \\ \hline &5&\color{blue}{10}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -47 } + \color{orangered}{ 40 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrr}4&5&-10&\color{orangered}{ -47 }\\& & 20& \color{orangered}{40} \\ \hline &\color{blue}{5}&\color{blue}{10}&\color{orangered}{-7} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x+10 } $ with a remainder of $ \color{red}{ -7 } $.