Divide using synthetic division $$ ( 5x^{2}+2x-8 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&5&2&-8\\& & -20& \color{black}{72} \\ \hline &\color{blue}{5}&\color{blue}{-18}&\color{orangered}{64} \end{array} $$The solution is:
$$ \frac{ 5x^{2}+2x-8 }{ x+4 } = \color{blue}{5x-18} ~+~ \frac{ \color{red}{ 64 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&2&-8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 5 }&2&-8\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&2&-8\\& & \color{blue}{-20} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrr}-4&5&\color{orangered}{ 2 }&-8\\& & \color{orangered}{-20} & \\ \hline &5&\color{orangered}{-18}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&2&-8\\& & -20& \color{blue}{72} \\ \hline &5&\color{blue}{-18}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 72 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrr}-4&5&2&\color{orangered}{ -8 }\\& & -20& \color{orangered}{72} \\ \hline &\color{blue}{5}&\color{blue}{-18}&\color{orangered}{64} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x-18 } $ with a remainder of $ \color{red}{ 64 } $.