Divide using synthetic division $$ ( 2x^{4}+4x^{2}-27x-19 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&2&0&4&-27&-19\\& & -6& 18& -66& \color{black}{279} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{22}&\color{blue}{-93}&\color{orangered}{260} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+4x^{2}-27x-19 }{ x+3 } = \color{blue}{2x^{3}-6x^{2}+22x-93} ~+~ \frac{ \color{red}{ 260 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&4&-27&-19\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 2 }&0&4&-27&-19\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&4&-27&-19\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-3&2&\color{orangered}{ 0 }&4&-27&-19\\& & \color{orangered}{-6} & & & \\ \hline &2&\color{orangered}{-6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&4&-27&-19\\& & -6& \color{blue}{18} & & \\ \hline &2&\color{blue}{-6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 18 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrr}-3&2&0&\color{orangered}{ 4 }&-27&-19\\& & -6& \color{orangered}{18} & & \\ \hline &2&-6&\color{orangered}{22}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 22 } = \color{blue}{ -66 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&4&-27&-19\\& & -6& 18& \color{blue}{-66} & \\ \hline &2&-6&\color{blue}{22}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ \left( -66 \right) } = \color{orangered}{ -93 } $
$$ \begin{array}{c|rrrrr}-3&2&0&4&\color{orangered}{ -27 }&-19\\& & -6& 18& \color{orangered}{-66} & \\ \hline &2&-6&22&\color{orangered}{-93}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -93 \right) } = \color{blue}{ 279 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&0&4&-27&-19\\& & -6& 18& -66& \color{blue}{279} \\ \hline &2&-6&22&\color{blue}{-93}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 279 } = \color{orangered}{ 260 } $
$$ \begin{array}{c|rrrrr}-3&2&0&4&-27&\color{orangered}{ -19 }\\& & -6& 18& -66& \color{orangered}{279} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{22}&\color{blue}{-93}&\color{orangered}{260} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-6x^{2}+22x-93 } $ with a remainder of $ \color{red}{ 260 } $.