Divide using synthetic division $$ ( 3x^{2}-x-2 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&3&-1&-2\\& & -12& \color{black}{52} \\ \hline &\color{blue}{3}&\color{blue}{-13}&\color{orangered}{50} \end{array} $$The solution is:
$$ \frac{ 3x^{2}-x-2 }{ x+4 } = \color{blue}{3x-13} ~+~ \frac{ \color{red}{ 50 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&3&-1&-2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 3 }&-1&-2\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&3&-1&-2\\& & \color{blue}{-12} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrr}-4&3&\color{orangered}{ -1 }&-2\\& & \color{orangered}{-12} & \\ \hline &3&\color{orangered}{-13}& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&3&-1&-2\\& & -12& \color{blue}{52} \\ \hline &3&\color{blue}{-13}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 52 } = \color{orangered}{ 50 } $
$$ \begin{array}{c|rrr}-4&3&-1&\color{orangered}{ -2 }\\& & -12& \color{orangered}{52} \\ \hline &\color{blue}{3}&\color{blue}{-13}&\color{orangered}{50} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-13 } $ with a remainder of $ \color{red}{ 50 } $.