Divide using synthetic division $$ ( -2x^{3}+5x^{2}-x+2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-2&5&-1&2\\& & 4& -18& \color{black}{38} \\ \hline &\color{blue}{-2}&\color{blue}{9}&\color{blue}{-19}&\color{orangered}{40} \end{array} $$The solution is:
$$ \frac{ -2x^{3}+5x^{2}-x+2 }{ x+2 } = \color{blue}{-2x^{2}+9x-19} ~+~ \frac{ \color{red}{ 40 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-2&5&-1&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ -2 }&5&-1&2\\& & & & \\ \hline &\color{orangered}{-2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-2&5&-1&2\\& & \color{blue}{4} & & \\ \hline &\color{blue}{-2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 4 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}-2&-2&\color{orangered}{ 5 }&-1&2\\& & \color{orangered}{4} & & \\ \hline &-2&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-2&5&-1&2\\& & 4& \color{blue}{-18} & \\ \hline &-2&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrr}-2&-2&5&\color{orangered}{ -1 }&2\\& & 4& \color{orangered}{-18} & \\ \hline &-2&9&\color{orangered}{-19}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 38 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-2&5&-1&2\\& & 4& -18& \color{blue}{38} \\ \hline &-2&9&\color{blue}{-19}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 38 } = \color{orangered}{ 40 } $
$$ \begin{array}{c|rrrr}-2&-2&5&-1&\color{orangered}{ 2 }\\& & 4& -18& \color{orangered}{38} \\ \hline &\color{blue}{-2}&\color{blue}{9}&\color{blue}{-19}&\color{orangered}{40} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{2}+9x-19 } $ with a remainder of $ \color{red}{ 40 } $.