Divide using synthetic division $$ ( 8x^{3}+3x^{2}+4x-7 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&8&3&4&-7\\& & -24& 63& \color{black}{-201} \\ \hline &\color{blue}{8}&\color{blue}{-21}&\color{blue}{67}&\color{orangered}{-208} \end{array} $$The solution is:
$$ \frac{ 8x^{3}+3x^{2}+4x-7 }{ x+3 } = \color{blue}{8x^{2}-21x+67} \color{red}{~-~} \frac{ \color{red}{ 208 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&3&4&-7\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 8 }&3&4&-7\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&3&4&-7\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrr}-3&8&\color{orangered}{ 3 }&4&-7\\& & \color{orangered}{-24} & & \\ \hline &8&\color{orangered}{-21}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -21 \right) } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&3&4&-7\\& & -24& \color{blue}{63} & \\ \hline &8&\color{blue}{-21}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 63 } = \color{orangered}{ 67 } $
$$ \begin{array}{c|rrrr}-3&8&3&\color{orangered}{ 4 }&-7\\& & -24& \color{orangered}{63} & \\ \hline &8&-21&\color{orangered}{67}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 67 } = \color{blue}{ -201 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&8&3&4&-7\\& & -24& 63& \color{blue}{-201} \\ \hline &8&-21&\color{blue}{67}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -201 \right) } = \color{orangered}{ -208 } $
$$ \begin{array}{c|rrrr}-3&8&3&4&\color{orangered}{ -7 }\\& & -24& 63& \color{orangered}{-201} \\ \hline &\color{blue}{8}&\color{blue}{-21}&\color{blue}{67}&\color{orangered}{-208} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-21x+67 } $ with a remainder of $ \color{red}{ -208 } $.