Divide using synthetic division $$ ( 7x^{4}-12x^{3}+2x-3 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&7&-12&0&2&-3\\& & -21& 99& -297& \color{black}{885} \\ \hline &\color{blue}{7}&\color{blue}{-33}&\color{blue}{99}&\color{blue}{-295}&\color{orangered}{882} \end{array} $$The solution is:
$$ \frac{ 7x^{4}-12x^{3}+2x-3 }{ x+3 } = \color{blue}{7x^{3}-33x^{2}+99x-295} ~+~ \frac{ \color{red}{ 882 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&7&-12&0&2&-3\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 7 }&-12&0&2&-3\\& & & & & \\ \hline &\color{orangered}{7}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 7 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&7&-12&0&2&-3\\& & \color{blue}{-21} & & & \\ \hline &\color{blue}{7}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -33 } $
$$ \begin{array}{c|rrrrr}-3&7&\color{orangered}{ -12 }&0&2&-3\\& & \color{orangered}{-21} & & & \\ \hline &7&\color{orangered}{-33}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -33 \right) } = \color{blue}{ 99 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&7&-12&0&2&-3\\& & -21& \color{blue}{99} & & \\ \hline &7&\color{blue}{-33}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 99 } = \color{orangered}{ 99 } $
$$ \begin{array}{c|rrrrr}-3&7&-12&\color{orangered}{ 0 }&2&-3\\& & -21& \color{orangered}{99} & & \\ \hline &7&-33&\color{orangered}{99}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 99 } = \color{blue}{ -297 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&7&-12&0&2&-3\\& & -21& 99& \color{blue}{-297} & \\ \hline &7&-33&\color{blue}{99}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -297 \right) } = \color{orangered}{ -295 } $
$$ \begin{array}{c|rrrrr}-3&7&-12&0&\color{orangered}{ 2 }&-3\\& & -21& 99& \color{orangered}{-297} & \\ \hline &7&-33&99&\color{orangered}{-295}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -295 \right) } = \color{blue}{ 885 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&7&-12&0&2&-3\\& & -21& 99& -297& \color{blue}{885} \\ \hline &7&-33&99&\color{blue}{-295}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 885 } = \color{orangered}{ 882 } $
$$ \begin{array}{c|rrrrr}-3&7&-12&0&2&\color{orangered}{ -3 }\\& & -21& 99& -297& \color{orangered}{885} \\ \hline &\color{blue}{7}&\color{blue}{-33}&\color{blue}{99}&\color{blue}{-295}&\color{orangered}{882} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x^{3}-33x^{2}+99x-295 } $ with a remainder of $ \color{red}{ 882 } $.