Divide using synthetic division $$ ( 5x^{2}+4x-9 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&5&4&-9\\& & -20& \color{black}{64} \\ \hline &\color{blue}{5}&\color{blue}{-16}&\color{orangered}{55} \end{array} $$The solution is:
$$ \frac{ 5x^{2}+4x-9 }{ x+4 } = \color{blue}{5x-16} ~+~ \frac{ \color{red}{ 55 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&4&-9\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 5 }&4&-9\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&4&-9\\& & \color{blue}{-20} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrr}-4&5&\color{orangered}{ 4 }&-9\\& & \color{orangered}{-20} & \\ \hline &5&\color{orangered}{-16}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&4&-9\\& & -20& \color{blue}{64} \\ \hline &5&\color{blue}{-16}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 64 } = \color{orangered}{ 55 } $
$$ \begin{array}{c|rrr}-4&5&4&\color{orangered}{ -9 }\\& & -20& \color{orangered}{64} \\ \hline &\color{blue}{5}&\color{blue}{-16}&\color{orangered}{55} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x-16 } $ with a remainder of $ \color{red}{ 55 } $.