Divide using synthetic division $$ ( -8x^{3}+3x^{2}+5 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-8&3&0&5\\& & 16& -38& \color{black}{76} \\ \hline &\color{blue}{-8}&\color{blue}{19}&\color{blue}{-38}&\color{orangered}{81} \end{array} $$The solution is:
$$ \frac{ -8x^{3}+3x^{2}+5 }{ x+2 } = \color{blue}{-8x^{2}+19x-38} ~+~ \frac{ \color{red}{ 81 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-8&3&0&5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ -8 }&3&0&5\\& & & & \\ \hline &\color{orangered}{-8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-8&3&0&5\\& & \color{blue}{16} & & \\ \hline &\color{blue}{-8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 16 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}-2&-8&\color{orangered}{ 3 }&0&5\\& & \color{orangered}{16} & & \\ \hline &-8&\color{orangered}{19}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 19 } = \color{blue}{ -38 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-8&3&0&5\\& & 16& \color{blue}{-38} & \\ \hline &-8&\color{blue}{19}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -38 \right) } = \color{orangered}{ -38 } $
$$ \begin{array}{c|rrrr}-2&-8&3&\color{orangered}{ 0 }&5\\& & 16& \color{orangered}{-38} & \\ \hline &-8&19&\color{orangered}{-38}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -38 \right) } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-8&3&0&5\\& & 16& -38& \color{blue}{76} \\ \hline &-8&19&\color{blue}{-38}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 76 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrr}-2&-8&3&0&\color{orangered}{ 5 }\\& & 16& -38& \color{orangered}{76} \\ \hline &\color{blue}{-8}&\color{blue}{19}&\color{blue}{-38}&\color{orangered}{81} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -8x^{2}+19x-38 } $ with a remainder of $ \color{red}{ 81 } $.