Divide using synthetic division $$ ( 8x^{2}+6x-20 ) \div ( 4x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&8&6&-20\\& & 40& \color{black}{230} \\ \hline &\color{blue}{8}&\color{blue}{46}&\color{orangered}{210} \end{array} $$The solution is:
$$ \frac{ 8x^{2}+6x-20 }{ x-5 } = \color{blue}{8x+46} ~+~ \frac{ \color{red}{ 210 } }{ x-5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{5}&8&6&-20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ 8 }&6&-20\\& & & \\ \hline &\color{orangered}{8}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 8 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&8&6&-20\\& & \color{blue}{40} & \\ \hline &\color{blue}{8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 40 } = \color{orangered}{ 46 } $
$$ \begin{array}{c|rrr}5&8&\color{orangered}{ 6 }&-20\\& & \color{orangered}{40} & \\ \hline &8&\color{orangered}{46}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 46 } = \color{blue}{ 230 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&8&6&-20\\& & 40& \color{blue}{230} \\ \hline &8&\color{blue}{46}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 230 } = \color{orangered}{ 210 } $
$$ \begin{array}{c|rrr}5&8&6&\color{orangered}{ -20 }\\& & 40& \color{orangered}{230} \\ \hline &\color{blue}{8}&\color{blue}{46}&\color{orangered}{210} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x+46 } $ with a remainder of $ \color{red}{ 210 } $.