Divide using synthetic division $$ ( -5x^{5}+7x^{4}-7x^{3}+2x-3 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&-5&7&-7&0&2&-3\\& & 15& -66& 219& -657& \color{black}{1965} \\ \hline &\color{blue}{-5}&\color{blue}{22}&\color{blue}{-73}&\color{blue}{219}&\color{blue}{-655}&\color{orangered}{1962} \end{array} $$The solution is:
$$ \frac{ -5x^{5}+7x^{4}-7x^{3}+2x-3 }{ x+3 } = \color{blue}{-5x^{4}+22x^{3}-73x^{2}+219x-655} ~+~ \frac{ \color{red}{ 1962 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-5&7&-7&0&2&-3\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ -5 }&7&-7&0&2&-3\\& & & & & & \\ \hline &\color{orangered}{-5}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-5&7&-7&0&2&-3\\& & \color{blue}{15} & & & & \\ \hline &\color{blue}{-5}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 15 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrrr}-3&-5&\color{orangered}{ 7 }&-7&0&2&-3\\& & \color{orangered}{15} & & & & \\ \hline &-5&\color{orangered}{22}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 22 } = \color{blue}{ -66 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-5&7&-7&0&2&-3\\& & 15& \color{blue}{-66} & & & \\ \hline &-5&\color{blue}{22}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -66 \right) } = \color{orangered}{ -73 } $
$$ \begin{array}{c|rrrrrr}-3&-5&7&\color{orangered}{ -7 }&0&2&-3\\& & 15& \color{orangered}{-66} & & & \\ \hline &-5&22&\color{orangered}{-73}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -73 \right) } = \color{blue}{ 219 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-5&7&-7&0&2&-3\\& & 15& -66& \color{blue}{219} & & \\ \hline &-5&22&\color{blue}{-73}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 219 } = \color{orangered}{ 219 } $
$$ \begin{array}{c|rrrrrr}-3&-5&7&-7&\color{orangered}{ 0 }&2&-3\\& & 15& -66& \color{orangered}{219} & & \\ \hline &-5&22&-73&\color{orangered}{219}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 219 } = \color{blue}{ -657 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-5&7&-7&0&2&-3\\& & 15& -66& 219& \color{blue}{-657} & \\ \hline &-5&22&-73&\color{blue}{219}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -657 \right) } = \color{orangered}{ -655 } $
$$ \begin{array}{c|rrrrrr}-3&-5&7&-7&0&\color{orangered}{ 2 }&-3\\& & 15& -66& 219& \color{orangered}{-657} & \\ \hline &-5&22&-73&219&\color{orangered}{-655}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -655 \right) } = \color{blue}{ 1965 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-5&7&-7&0&2&-3\\& & 15& -66& 219& -657& \color{blue}{1965} \\ \hline &-5&22&-73&219&\color{blue}{-655}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 1965 } = \color{orangered}{ 1962 } $
$$ \begin{array}{c|rrrrrr}-3&-5&7&-7&0&2&\color{orangered}{ -3 }\\& & 15& -66& 219& -657& \color{orangered}{1965} \\ \hline &\color{blue}{-5}&\color{blue}{22}&\color{blue}{-73}&\color{blue}{219}&\color{blue}{-655}&\color{orangered}{1962} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{4}+22x^{3}-73x^{2}+219x-655 } $ with a remainder of $ \color{red}{ 1962 } $.