Divide using synthetic division $$ ( -5x^{3}-8x^{2}+8 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-5&-8&0&8\\& & 10& -4& \color{black}{8} \\ \hline &\color{blue}{-5}&\color{blue}{2}&\color{blue}{-4}&\color{orangered}{16} \end{array} $$The solution is:
$$ \frac{ -5x^{3}-8x^{2}+8 }{ x+2 } = \color{blue}{-5x^{2}+2x-4} ~+~ \frac{ \color{red}{ 16 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-8&0&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ -5 }&-8&0&8\\& & & & \\ \hline &\color{orangered}{-5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-8&0&8\\& & \color{blue}{10} & & \\ \hline &\color{blue}{-5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 10 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-2&-5&\color{orangered}{ -8 }&0&8\\& & \color{orangered}{10} & & \\ \hline &-5&\color{orangered}{2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-8&0&8\\& & 10& \color{blue}{-4} & \\ \hline &-5&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-2&-5&-8&\color{orangered}{ 0 }&8\\& & 10& \color{orangered}{-4} & \\ \hline &-5&2&\color{orangered}{-4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-8&0&8\\& & 10& -4& \color{blue}{8} \\ \hline &-5&2&\color{blue}{-4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 8 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrr}-2&-5&-8&0&\color{orangered}{ 8 }\\& & 10& -4& \color{orangered}{8} \\ \hline &\color{blue}{-5}&\color{blue}{2}&\color{blue}{-4}&\color{orangered}{16} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{2}+2x-4 } $ with a remainder of $ \color{red}{ 16 } $.