Divide using synthetic division $$ ( 5x^{3}-3x^{2}+3x-4 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&5&-3&3&-4\\& & -15& 54& \color{black}{-171} \\ \hline &\color{blue}{5}&\color{blue}{-18}&\color{blue}{57}&\color{orangered}{-175} \end{array} $$The solution is:
$$ \frac{ 5x^{3}-3x^{2}+3x-4 }{ x+3 } = \color{blue}{5x^{2}-18x+57} \color{red}{~-~} \frac{ \color{red}{ 175 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&5&-3&3&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 5 }&-3&3&-4\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 5 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&5&-3&3&-4\\& & \color{blue}{-15} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrr}-3&5&\color{orangered}{ -3 }&3&-4\\& & \color{orangered}{-15} & & \\ \hline &5&\color{orangered}{-18}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&5&-3&3&-4\\& & -15& \color{blue}{54} & \\ \hline &5&\color{blue}{-18}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 54 } = \color{orangered}{ 57 } $
$$ \begin{array}{c|rrrr}-3&5&-3&\color{orangered}{ 3 }&-4\\& & -15& \color{orangered}{54} & \\ \hline &5&-18&\color{orangered}{57}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 57 } = \color{blue}{ -171 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&5&-3&3&-4\\& & -15& 54& \color{blue}{-171} \\ \hline &5&-18&\color{blue}{57}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -171 \right) } = \color{orangered}{ -175 } $
$$ \begin{array}{c|rrrr}-3&5&-3&3&\color{orangered}{ -4 }\\& & -15& 54& \color{orangered}{-171} \\ \hline &\color{blue}{5}&\color{blue}{-18}&\color{blue}{57}&\color{orangered}{-175} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}-18x+57 } $ with a remainder of $ \color{red}{ -175 } $.