Divide using synthetic division $$ ( 3x^{3}-x^{2}+x-2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&3&-1&1&-2\\& & -6& 14& \color{black}{-30} \\ \hline &\color{blue}{3}&\color{blue}{-7}&\color{blue}{15}&\color{orangered}{-32} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-x^{2}+x-2 }{ x+2 } = \color{blue}{3x^{2}-7x+15} \color{red}{~-~} \frac{ \color{red}{ 32 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-1&1&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 3 }&-1&1&-2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-1&1&-2\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}-2&3&\color{orangered}{ -1 }&1&-2\\& & \color{orangered}{-6} & & \\ \hline &3&\color{orangered}{-7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-1&1&-2\\& & -6& \color{blue}{14} & \\ \hline &3&\color{blue}{-7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 14 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrr}-2&3&-1&\color{orangered}{ 1 }&-2\\& & -6& \color{orangered}{14} & \\ \hline &3&-7&\color{orangered}{15}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 15 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-1&1&-2\\& & -6& 14& \color{blue}{-30} \\ \hline &3&-7&\color{blue}{15}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrrr}-2&3&-1&1&\color{orangered}{ -2 }\\& & -6& 14& \color{orangered}{-30} \\ \hline &\color{blue}{3}&\color{blue}{-7}&\color{blue}{15}&\color{orangered}{-32} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-7x+15 } $ with a remainder of $ \color{red}{ -32 } $.