Divide using synthetic division $$ ( x^{2}-9x+8 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&1&-9&8\\& & -4& \color{black}{52} \\ \hline &\color{blue}{1}&\color{blue}{-13}&\color{orangered}{60} \end{array} $$The solution is:
$$ \frac{ x^{2}-9x+8 }{ x+4 } = \color{blue}{x-13} ~+~ \frac{ \color{red}{ 60 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-9&8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 1 }&-9&8\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-9&8\\& & \color{blue}{-4} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrr}-4&1&\color{orangered}{ -9 }&8\\& & \color{orangered}{-4} & \\ \hline &1&\color{orangered}{-13}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 52 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-9&8\\& & -4& \color{blue}{52} \\ \hline &1&\color{blue}{-13}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 52 } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrr}-4&1&-9&\color{orangered}{ 8 }\\& & -4& \color{orangered}{52} \\ \hline &\color{blue}{1}&\color{blue}{-13}&\color{orangered}{60} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-13 } $ with a remainder of $ \color{red}{ 60 } $.