Check if $ x+4 $ is a factor of $ x^{3}-5x^{2}-45x-46 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-4&1&-5&-45&-46\\& & -4& 36& \color{black}{36} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{-9}&\color{orangered}{-10} \end{array} $$

Because the remainder $ \left( \color{red}{ -10 } \right) $ is not zero, we conclude that the $ x+4 $ is not a factor of $ x^{3}-5x^{2}-45x-46$.

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 + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-5&-45&-46\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 1 }&-5&-45&-46\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-5&-45&-46\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -9 } $

$$ \begin{array}{c|rrrr}-4&1&\color{orangered}{ -5 }&-45&-46\\& & \color{orangered}{-4} & & \\ \hline &1&\color{orangered}{-9}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 36 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-5&-45&-46\\& & -4& \color{blue}{36} & \\ \hline &1&\color{blue}{-9}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -45 } + \color{orangered}{ 36 } = \color{orangered}{ -9 } $

$$ \begin{array}{c|rrrr}-4&1&-5&\color{orangered}{ -45 }&-46\\& & -4& \color{orangered}{36} & \\ \hline &1&-9&\color{orangered}{-9}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 36 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-5&-45&-46\\& & -4& 36& \color{blue}{36} \\ \hline &1&-9&\color{blue}{-9}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -46 } + \color{orangered}{ 36 } = \color{orangered}{ -10 } $

$$ \begin{array}{c|rrrr}-4&1&-5&-45&\color{orangered}{ -46 }\\& & -4& 36& \color{orangered}{36} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{-9}&\color{orangered}{-10} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -10 }\right)$.

Synthetic Division Calculator