Divide using synthetic division $$ ( 3x^{2}-2x+20 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&3&-2&20\\& & -12& \color{black}{56} \\ \hline &\color{blue}{3}&\color{blue}{-14}&\color{orangered}{76} \end{array} $$The solution is:
$$ \frac{ 3x^{2}-2x+20 }{ x+4 } = \color{blue}{3x-14} ~+~ \frac{ \color{red}{ 76 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&3&-2&20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 3 }&-2&20\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 3 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&3&-2&20\\& & \color{blue}{-12} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrr}-4&3&\color{orangered}{ -2 }&20\\& & \color{orangered}{-12} & \\ \hline &3&\color{orangered}{-14}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 56 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&3&-2&20\\& & -12& \color{blue}{56} \\ \hline &3&\color{blue}{-14}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ 56 } = \color{orangered}{ 76 } $
$$ \begin{array}{c|rrr}-4&3&-2&\color{orangered}{ 20 }\\& & -12& \color{orangered}{56} \\ \hline &\color{blue}{3}&\color{blue}{-14}&\color{orangered}{76} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-14 } $ with a remainder of $ \color{red}{ 76 } $.