Check if $ x+3 $ is a factor of $ -2x^{3}-4x^{2}-6 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-3&-2&-4&0&-6\\& & 6& -6& \color{black}{18} \\ \hline &\color{blue}{-2}&\color{blue}{2}&\color{blue}{-6}&\color{orangered}{12} \end{array} $$

Because the remainder $ \left( \color{red}{ 12 } \right) $ is not zero, we conclude that the $ x+3 $ is not a factor of $ -2x^{3}-4x^{2}-6$.

First we need to create a synthetic division table.

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.

$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&-4&0&-6\\& & & & \\ \hline &&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrr}-3&\color{orangered}{ -2 }&-4&0&-6\\& & & & \\ \hline &\color{orangered}{-2}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 6 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&-4&0&-6\\& & \color{blue}{6} & & \\ \hline &\color{blue}{-2}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 6 } = \color{orangered}{ 2 } $

$$ \begin{array}{c|rrrr}-3&-2&\color{orangered}{ -4 }&0&-6\\& & \color{orangered}{6} & & \\ \hline &-2&\color{orangered}{2}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&-4&0&-6\\& & 6& \color{blue}{-6} & \\ \hline &-2&\color{blue}{2}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $

$$ \begin{array}{c|rrrr}-3&-2&-4&\color{orangered}{ 0 }&-6\\& & 6& \color{orangered}{-6} & \\ \hline &-2&2&\color{orangered}{-6}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&-4&0&-6\\& & 6& -6& \color{blue}{18} \\ \hline &-2&2&\color{blue}{-6}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 18 } = \color{orangered}{ 12 } $

$$ \begin{array}{c|rrrr}-3&-2&-4&0&\color{orangered}{ -6 }\\& & 6& -6& \color{orangered}{18} \\ \hline &\color{blue}{-2}&\color{blue}{2}&\color{blue}{-6}&\color{orangered}{12} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 12 }\right)$.

Synthetic Division Calculator