Divide using synthetic division $$ ( -3x^{4}+2x^{3}-2x^{2}+2 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&-3&2&-2&0&2\\& & -6& -8& -20& \color{black}{-40} \\ \hline &\color{blue}{-3}&\color{blue}{-4}&\color{blue}{-10}&\color{blue}{-20}&\color{orangered}{-38} \end{array} $$The solution is:
$$ \frac{ -3x^{4}+2x^{3}-2x^{2}+2 }{ x-2 } = \color{blue}{-3x^{3}-4x^{2}-10x-20} \color{red}{~-~} \frac{ \color{red}{ 38 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-3&2&-2&0&2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ -3 }&2&-2&0&2\\& & & & & \\ \hline &\color{orangered}{-3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-3&2&-2&0&2\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{-3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}2&-3&\color{orangered}{ 2 }&-2&0&2\\& & \color{orangered}{-6} & & & \\ \hline &-3&\color{orangered}{-4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-3&2&-2&0&2\\& & -6& \color{blue}{-8} & & \\ \hline &-3&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}2&-3&2&\color{orangered}{ -2 }&0&2\\& & -6& \color{orangered}{-8} & & \\ \hline &-3&-4&\color{orangered}{-10}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-3&2&-2&0&2\\& & -6& -8& \color{blue}{-20} & \\ \hline &-3&-4&\color{blue}{-10}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrrr}2&-3&2&-2&\color{orangered}{ 0 }&2\\& & -6& -8& \color{orangered}{-20} & \\ \hline &-3&-4&-10&\color{orangered}{-20}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-3&2&-2&0&2\\& & -6& -8& -20& \color{blue}{-40} \\ \hline &-3&-4&-10&\color{blue}{-20}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -38 } $
$$ \begin{array}{c|rrrrr}2&-3&2&-2&0&\color{orangered}{ 2 }\\& & -6& -8& -20& \color{orangered}{-40} \\ \hline &\color{blue}{-3}&\color{blue}{-4}&\color{blue}{-10}&\color{blue}{-20}&\color{orangered}{-38} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{3}-4x^{2}-10x-20 } $ with a remainder of $ \color{red}{ -38 } $.