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