Divide using synthetic division $$ ( 3x^{3}-13x^{2}+7x-2 ) \div ( x+3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&3&-13&7&-2\\& & -9& 66& \color{black}{-219} \\ \hline &\color{blue}{3}&\color{blue}{-22}&\color{blue}{73}&\color{orangered}{-221} \end{array} $$The solution is:
$$ \dfrac{ 3x^{3}-13x^{2}+7x-2 }{ x+3 } = \color{blue}{3x^{2}-22x+73} \color{red}{~-~} \dfrac{ \color{red}{ 221 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&-13&7&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 3 }&-13&7&-2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&-13&7&-2\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrrr}-3&3&\color{orangered}{ -13 }&7&-2\\& & \color{orangered}{-9} & & \\ \hline &3&\color{orangered}{-22}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ 66 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&-13&7&-2\\& & -9& \color{blue}{66} & \\ \hline &3&\color{blue}{-22}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 66 } = \color{orangered}{ 73 } $
$$ \begin{array}{c|rrrr}-3&3&-13&\color{orangered}{ 7 }&-2\\& & -9& \color{orangered}{66} & \\ \hline &3&-22&\color{orangered}{73}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 73 } = \color{blue}{ -219 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&-13&7&-2\\& & -9& 66& \color{blue}{-219} \\ \hline &3&-22&\color{blue}{73}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -219 \right) } = \color{orangered}{ -221 } $
$$ \begin{array}{c|rrrr}-3&3&-13&7&\color{orangered}{ -2 }\\& & -9& 66& \color{orangered}{-219} \\ \hline &\color{blue}{3}&\color{blue}{-22}&\color{blue}{73}&\color{orangered}{-221} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-22x+73 } $ with a remainder of $ \color{red}{ -221 } $.