Divide using synthetic division $$ ( 6x^{2}+7x-20 ) \div ( 2x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&6&7&-20\\& & -30& \color{black}{115} \\ \hline &\color{blue}{6}&\color{blue}{-23}&\color{orangered}{95} \end{array} $$The solution is:
$$ \frac{ 6x^{2}+7x-20 }{ x+5 } = \color{blue}{6x-23} ~+~ \frac{ \color{red}{ 95 } }{ x+5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-5}&6&7&-20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 6 }&7&-20\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 6 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&6&7&-20\\& & \color{blue}{-30} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrr}-5&6&\color{orangered}{ 7 }&-20\\& & \color{orangered}{-30} & \\ \hline &6&\color{orangered}{-23}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ 115 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&6&7&-20\\& & -30& \color{blue}{115} \\ \hline &6&\color{blue}{-23}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 115 } = \color{orangered}{ 95 } $
$$ \begin{array}{c|rrr}-5&6&7&\color{orangered}{ -20 }\\& & -30& \color{orangered}{115} \\ \hline &\color{blue}{6}&\color{blue}{-23}&\color{orangered}{95} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-23 } $ with a remainder of $ \color{red}{ 95 } $.