Divide using synthetic division $$ ( 3x^{4}-7x^{3}+9 ) \div ( x+3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&3&-7&0&0&9\\& & -9& 48& -144& \color{black}{432} \\ \hline &\color{blue}{3}&\color{blue}{-16}&\color{blue}{48}&\color{blue}{-144}&\color{orangered}{441} \end{array} $$The solution is:
$$ \dfrac{ 3x^{4}-7x^{3}+9 }{ x+3 } = \color{blue}{3x^{3}-16x^{2}+48x-144} ~+~ \dfrac{ \color{red}{ 441 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&-7&0&0&9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 3 }&-7&0&0&9\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&-7&0&0&9\\& & \color{blue}{-9} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrr}-3&3&\color{orangered}{ -7 }&0&0&9\\& & \color{orangered}{-9} & & & \\ \hline &3&\color{orangered}{-16}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&-7&0&0&9\\& & -9& \color{blue}{48} & & \\ \hline &3&\color{blue}{-16}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 48 } = \color{orangered}{ 48 } $
$$ \begin{array}{c|rrrrr}-3&3&-7&\color{orangered}{ 0 }&0&9\\& & -9& \color{orangered}{48} & & \\ \hline &3&-16&\color{orangered}{48}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 48 } = \color{blue}{ -144 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&-7&0&0&9\\& & -9& 48& \color{blue}{-144} & \\ \hline &3&-16&\color{blue}{48}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -144 \right) } = \color{orangered}{ -144 } $
$$ \begin{array}{c|rrrrr}-3&3&-7&0&\color{orangered}{ 0 }&9\\& & -9& 48& \color{orangered}{-144} & \\ \hline &3&-16&48&\color{orangered}{-144}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -144 \right) } = \color{blue}{ 432 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&3&-7&0&0&9\\& & -9& 48& -144& \color{blue}{432} \\ \hline &3&-16&48&\color{blue}{-144}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 432 } = \color{orangered}{ 441 } $
$$ \begin{array}{c|rrrrr}-3&3&-7&0&0&\color{orangered}{ 9 }\\& & -9& 48& -144& \color{orangered}{432} \\ \hline &\color{blue}{3}&\color{blue}{-16}&\color{blue}{48}&\color{blue}{-144}&\color{orangered}{441} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-16x^{2}+48x-144 } $ with a remainder of $ \color{red}{ 441 } $.