Divide using synthetic division $$ ( -2x^{2}-5x+7 ) \div ( x+8 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}-8&-2&-5&7\\& & 16& \color{black}{-88} \\ \hline &\color{blue}{-2}&\color{blue}{11}&\color{orangered}{-81} \end{array} $$The solution is:
$$ \dfrac{ -2x^{2}-5x+7 }{ x+8 } = \color{blue}{-2x+11} \color{red}{~-~} \dfrac{ \color{red}{ 81 } }{ x+8 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 8 = 0 $ ( $ x = \color{blue}{ -8 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-8}&-2&-5&7\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-8&\color{orangered}{ -2 }&-5&7\\& & & \\ \hline &\color{orangered}{-2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&-2&-5&7\\& & \color{blue}{16} & \\ \hline &\color{blue}{-2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 16 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrr}-8&-2&\color{orangered}{ -5 }&7\\& & \color{orangered}{16} & \\ \hline &-2&\color{orangered}{11}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 11 } = \color{blue}{ -88 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&-2&-5&7\\& & 16& \color{blue}{-88} \\ \hline &-2&\color{blue}{11}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -88 \right) } = \color{orangered}{ -81 } $
$$ \begin{array}{c|rrr}-8&-2&-5&\color{orangered}{ 7 }\\& & 16& \color{orangered}{-88} \\ \hline &\color{blue}{-2}&\color{blue}{11}&\color{orangered}{-81} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x+11 } $ with a remainder of $ \color{red}{ -81 } $.