Divide using synthetic division $$ ( -4x^{5}-6x^{3}+2x^{2}+4x-8 ) \div ( 9x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&-4&0&-6&2&4&-8\\& & 12& -36& 126& -384& \color{black}{1140} \\ \hline &\color{blue}{-4}&\color{blue}{12}&\color{blue}{-42}&\color{blue}{128}&\color{blue}{-380}&\color{orangered}{1132} \end{array} $$The solution is:
$$ \frac{ -4x^{5}-6x^{3}+2x^{2}+4x-8 }{ x+3 } = \color{blue}{-4x^{4}+12x^{3}-42x^{2}+128x-380} ~+~ \frac{ \color{red}{ 1132 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-4&0&-6&2&4&-8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ -4 }&0&-6&2&4&-8\\& & & & & & \\ \hline &\color{orangered}{-4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-4&0&-6&2&4&-8\\& & \color{blue}{12} & & & & \\ \hline &\color{blue}{-4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrr}-3&-4&\color{orangered}{ 0 }&-6&2&4&-8\\& & \color{orangered}{12} & & & & \\ \hline &-4&\color{orangered}{12}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 12 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-4&0&-6&2&4&-8\\& & 12& \color{blue}{-36} & & & \\ \hline &-4&\color{blue}{12}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -42 } $
$$ \begin{array}{c|rrrrrr}-3&-4&0&\color{orangered}{ -6 }&2&4&-8\\& & 12& \color{orangered}{-36} & & & \\ \hline &-4&12&\color{orangered}{-42}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -42 \right) } = \color{blue}{ 126 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-4&0&-6&2&4&-8\\& & 12& -36& \color{blue}{126} & & \\ \hline &-4&12&\color{blue}{-42}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 126 } = \color{orangered}{ 128 } $
$$ \begin{array}{c|rrrrrr}-3&-4&0&-6&\color{orangered}{ 2 }&4&-8\\& & 12& -36& \color{orangered}{126} & & \\ \hline &-4&12&-42&\color{orangered}{128}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 128 } = \color{blue}{ -384 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-4&0&-6&2&4&-8\\& & 12& -36& 126& \color{blue}{-384} & \\ \hline &-4&12&-42&\color{blue}{128}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -384 \right) } = \color{orangered}{ -380 } $
$$ \begin{array}{c|rrrrrr}-3&-4&0&-6&2&\color{orangered}{ 4 }&-8\\& & 12& -36& 126& \color{orangered}{-384} & \\ \hline &-4&12&-42&128&\color{orangered}{-380}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -380 \right) } = \color{blue}{ 1140 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-4&0&-6&2&4&-8\\& & 12& -36& 126& -384& \color{blue}{1140} \\ \hline &-4&12&-42&128&\color{blue}{-380}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 1140 } = \color{orangered}{ 1132 } $
$$ \begin{array}{c|rrrrrr}-3&-4&0&-6&2&4&\color{orangered}{ -8 }\\& & 12& -36& 126& -384& \color{orangered}{1140} \\ \hline &\color{blue}{-4}&\color{blue}{12}&\color{blue}{-42}&\color{blue}{128}&\color{blue}{-380}&\color{orangered}{1132} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{4}+12x^{3}-42x^{2}+128x-380 } $ with a remainder of $ \color{red}{ 1132 } $.