Divide using synthetic division $$ ( -4x^{3}+3x^{2}-x+3 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-4&3&-1&3\\& & 8& -22& \color{black}{46} \\ \hline &\color{blue}{-4}&\color{blue}{11}&\color{blue}{-23}&\color{orangered}{49} \end{array} $$The solution is:
$$ \frac{ -4x^{3}+3x^{2}-x+3 }{ x+2 } = \color{blue}{-4x^{2}+11x-23} ~+~ \frac{ \color{red}{ 49 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&3&-1&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ -4 }&3&-1&3\\& & & & \\ \hline &\color{orangered}{-4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&3&-1&3\\& & \color{blue}{8} & & \\ \hline &\color{blue}{-4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 8 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}-2&-4&\color{orangered}{ 3 }&-1&3\\& & \color{orangered}{8} & & \\ \hline &-4&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 11 } = \color{blue}{ -22 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&3&-1&3\\& & 8& \color{blue}{-22} & \\ \hline &-4&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -22 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrrr}-2&-4&3&\color{orangered}{ -1 }&3\\& & 8& \color{orangered}{-22} & \\ \hline &-4&11&\color{orangered}{-23}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ 46 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&3&-1&3\\& & 8& -22& \color{blue}{46} \\ \hline &-4&11&\color{blue}{-23}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 46 } = \color{orangered}{ 49 } $
$$ \begin{array}{c|rrrr}-2&-4&3&-1&\color{orangered}{ 3 }\\& & 8& -22& \color{orangered}{46} \\ \hline &\color{blue}{-4}&\color{blue}{11}&\color{blue}{-23}&\color{orangered}{49} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{2}+11x-23 } $ with a remainder of $ \color{red}{ 49 } $.