Divide using synthetic division $$ ( 2x^{5}-3x^{3}-14x ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&2&0&-3&0&-14&0\\& & -6& 18& -45& 135& \color{black}{-363} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{15}&\color{blue}{-45}&\color{blue}{121}&\color{orangered}{-363} \end{array} $$The solution is:
$$ \frac{ 2x^{5}-3x^{3}-14x }{ x+3 } = \color{blue}{2x^{4}-6x^{3}+15x^{2}-45x+121} \color{red}{~-~} \frac{ \color{red}{ 363 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&0&-3&0&-14&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 2 }&0&-3&0&-14&0\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&0&-3&0&-14&0\\& & \color{blue}{-6} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrr}-3&2&\color{orangered}{ 0 }&-3&0&-14&0\\& & \color{orangered}{-6} & & & & \\ \hline &2&\color{orangered}{-6}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&0&-3&0&-14&0\\& & -6& \color{blue}{18} & & & \\ \hline &2&\color{blue}{-6}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 18 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrrr}-3&2&0&\color{orangered}{ -3 }&0&-14&0\\& & -6& \color{orangered}{18} & & & \\ \hline &2&-6&\color{orangered}{15}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 15 } = \color{blue}{ -45 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&0&-3&0&-14&0\\& & -6& 18& \color{blue}{-45} & & \\ \hline &2&-6&\color{blue}{15}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -45 \right) } = \color{orangered}{ -45 } $
$$ \begin{array}{c|rrrrrr}-3&2&0&-3&\color{orangered}{ 0 }&-14&0\\& & -6& 18& \color{orangered}{-45} & & \\ \hline &2&-6&15&\color{orangered}{-45}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -45 \right) } = \color{blue}{ 135 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&0&-3&0&-14&0\\& & -6& 18& -45& \color{blue}{135} & \\ \hline &2&-6&15&\color{blue}{-45}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 135 } = \color{orangered}{ 121 } $
$$ \begin{array}{c|rrrrrr}-3&2&0&-3&0&\color{orangered}{ -14 }&0\\& & -6& 18& -45& \color{orangered}{135} & \\ \hline &2&-6&15&-45&\color{orangered}{121}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 121 } = \color{blue}{ -363 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&0&-3&0&-14&0\\& & -6& 18& -45& 135& \color{blue}{-363} \\ \hline &2&-6&15&-45&\color{blue}{121}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -363 \right) } = \color{orangered}{ -363 } $
$$ \begin{array}{c|rrrrrr}-3&2&0&-3&0&-14&\color{orangered}{ 0 }\\& & -6& 18& -45& 135& \color{orangered}{-363} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{15}&\color{blue}{-45}&\color{blue}{121}&\color{orangered}{-363} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{4}-6x^{3}+15x^{2}-45x+121 } $ with a remainder of $ \color{red}{ -363 } $.