Divide using synthetic division $$ ( 4x^{2}-13x+8 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&4&-13&8\\& & -20& \color{black}{165} \\ \hline &\color{blue}{4}&\color{blue}{-33}&\color{orangered}{173} \end{array} $$The solution is:
$$ \frac{ 4x^{2}-13x+8 }{ x+5 } = \color{blue}{4x-33} ~+~ \frac{ \color{red}{ 173 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-5}&4&-13&8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 4 }&-13&8\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&4&-13&8\\& & \color{blue}{-20} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -33 } $
$$ \begin{array}{c|rrr}-5&4&\color{orangered}{ -13 }&8\\& & \color{orangered}{-20} & \\ \hline &4&\color{orangered}{-33}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -33 \right) } = \color{blue}{ 165 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&4&-13&8\\& & -20& \color{blue}{165} \\ \hline &4&\color{blue}{-33}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 165 } = \color{orangered}{ 173 } $
$$ \begin{array}{c|rrr}-5&4&-13&\color{orangered}{ 8 }\\& & -20& \color{orangered}{165} \\ \hline &\color{blue}{4}&\color{blue}{-33}&\color{orangered}{173} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x-33 } $ with a remainder of $ \color{red}{ 173 } $.