Divide using synthetic division $$ ( -3x^{4}+2x^{3}+3x^{2}-4x+5 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&-3&2&3&-4&5\\& & 6& -16& 26& \color{black}{-44} \\ \hline &\color{blue}{-3}&\color{blue}{8}&\color{blue}{-13}&\color{blue}{22}&\color{orangered}{-39} \end{array} $$The solution is:
$$ \frac{ -3x^{4}+2x^{3}+3x^{2}-4x+5 }{ x+2 } = \color{blue}{-3x^{3}+8x^{2}-13x+22} \color{red}{~-~} \frac{ \color{red}{ 39 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-3&2&3&-4&5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ -3 }&2&3&-4&5\\& & & & & \\ \hline &\color{orangered}{-3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-3&2&3&-4&5\\& & \color{blue}{6} & & & \\ \hline &\color{blue}{-3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 6 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}-2&-3&\color{orangered}{ 2 }&3&-4&5\\& & \color{orangered}{6} & & & \\ \hline &-3&\color{orangered}{8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-3&2&3&-4&5\\& & 6& \color{blue}{-16} & & \\ \hline &-3&\color{blue}{8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrrr}-2&-3&2&\color{orangered}{ 3 }&-4&5\\& & 6& \color{orangered}{-16} & & \\ \hline &-3&8&\color{orangered}{-13}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-3&2&3&-4&5\\& & 6& -16& \color{blue}{26} & \\ \hline &-3&8&\color{blue}{-13}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 26 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrr}-2&-3&2&3&\color{orangered}{ -4 }&5\\& & 6& -16& \color{orangered}{26} & \\ \hline &-3&8&-13&\color{orangered}{22}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 22 } = \color{blue}{ -44 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-3&2&3&-4&5\\& & 6& -16& 26& \color{blue}{-44} \\ \hline &-3&8&-13&\color{blue}{22}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -44 \right) } = \color{orangered}{ -39 } $
$$ \begin{array}{c|rrrrr}-2&-3&2&3&-4&\color{orangered}{ 5 }\\& & 6& -16& 26& \color{orangered}{-44} \\ \hline &\color{blue}{-3}&\color{blue}{8}&\color{blue}{-13}&\color{blue}{22}&\color{orangered}{-39} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{3}+8x^{2}-13x+22 } $ with a remainder of $ \color{red}{ -39 } $.