Divide using synthetic division $$ ( 8x^{3}-4x^{2}-7x-6 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&8&-4&-7&-6\\& & -24& 84& \color{black}{-231} \\ \hline &\color{blue}{8}&\color{blue}{-28}&\color{blue}{77}&\color{orangered}{-237} \end{array} $$The solution is:
$$ \frac{ 8x^{3}-4x^{2}-7x-6 }{ x+3 } = \color{blue}{8x^{2}-28x+77} \color{red}{~-~} \frac{ \color{red}{ 237 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&-4&-7&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 8 }&-4&-7&-6\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 8 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&-4&-7&-6\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -28 } $
$$ \begin{array}{c|rrrr}-3&8&\color{orangered}{ -4 }&-7&-6\\& & \color{orangered}{-24} & & \\ \hline &8&\color{orangered}{-28}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -28 \right) } = \color{blue}{ 84 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&-4&-7&-6\\& & -24& \color{blue}{84} & \\ \hline &8&\color{blue}{-28}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 84 } = \color{orangered}{ 77 } $
$$ \begin{array}{c|rrrr}-3&8&-4&\color{orangered}{ -7 }&-6\\& & -24& \color{orangered}{84} & \\ \hline &8&-28&\color{orangered}{77}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 77 } = \color{blue}{ -231 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&-4&-7&-6\\& & -24& 84& \color{blue}{-231} \\ \hline &8&-28&\color{blue}{77}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -231 \right) } = \color{orangered}{ -237 } $
$$ \begin{array}{c|rrrr}-3&8&-4&-7&\color{orangered}{ -6 }\\& & -24& 84& \color{orangered}{-231} \\ \hline &\color{blue}{8}&\color{blue}{-28}&\color{blue}{77}&\color{orangered}{-237} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-28x+77 } $ with a remainder of $ \color{red}{ -237 } $.