Divide using synthetic division $$ ( -4x^{3}+2x^{2}-x+1 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-4&2&-1&1\\& & 8& -20& \color{black}{42} \\ \hline &\color{blue}{-4}&\color{blue}{10}&\color{blue}{-21}&\color{orangered}{43} \end{array} $$The solution is:
$$ \frac{ -4x^{3}+2x^{2}-x+1 }{ x+2 } = \color{blue}{-4x^{2}+10x-21} ~+~ \frac{ \color{red}{ 43 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&2&-1&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ -4 }&2&-1&1\\& & & & \\ \hline &\color{orangered}{-4}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&2&-1&1\\& & \color{blue}{8} & & \\ \hline &\color{blue}{-4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 8 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}-2&-4&\color{orangered}{ 2 }&-1&1\\& & \color{orangered}{8} & & \\ \hline &-4&\color{orangered}{10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 10 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&2&-1&1\\& & 8& \color{blue}{-20} & \\ \hline &-4&\color{blue}{10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrr}-2&-4&2&\color{orangered}{ -1 }&1\\& & 8& \color{orangered}{-20} & \\ \hline &-4&10&\color{orangered}{-21}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -21 \right) } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-4&2&-1&1\\& & 8& -20& \color{blue}{42} \\ \hline &-4&10&\color{blue}{-21}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 42 } = \color{orangered}{ 43 } $
$$ \begin{array}{c|rrrr}-2&-4&2&-1&\color{orangered}{ 1 }\\& & 8& -20& \color{orangered}{42} \\ \hline &\color{blue}{-4}&\color{blue}{10}&\color{blue}{-21}&\color{orangered}{43} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{2}+10x-21 } $ with a remainder of $ \color{red}{ 43 } $.