Divide using synthetic division $$ ( x^{3}-4x^{2}+2x-3 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&1&-4&2&-3\\& & -2& 12& \color{black}{-28} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{14}&\color{orangered}{-31} \end{array} $$The solution is:
$$ \frac{ x^{3}-4x^{2}+2x-3 }{ x+2 } = \color{blue}{x^{2}-6x+14} \color{red}{~-~} \frac{ \color{red}{ 31 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-4&2&-3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 1 }&-4&2&-3\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-4&2&-3\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-2&1&\color{orangered}{ -4 }&2&-3\\& & \color{orangered}{-2} & & \\ \hline &1&\color{orangered}{-6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-4&2&-3\\& & -2& \color{blue}{12} & \\ \hline &1&\color{blue}{-6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 12 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}-2&1&-4&\color{orangered}{ 2 }&-3\\& & -2& \color{orangered}{12} & \\ \hline &1&-6&\color{orangered}{14}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 14 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-4&2&-3\\& & -2& 12& \color{blue}{-28} \\ \hline &1&-6&\color{blue}{14}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrrr}-2&1&-4&2&\color{orangered}{ -3 }\\& & -2& 12& \color{orangered}{-28} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{14}&\color{orangered}{-31} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-6x+14 } $ with a remainder of $ \color{red}{ -31 } $.