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

The synthetic division table is:

$$ \begin{array}{c|rrrr}-4&1&5&5&4\\& & -4& -4& \color{black}{-4} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{orangered}{0} \end{array} $$

Because the remainder equals zero, we conclude that the $ x+4 $ is a factor of the $ x^{3}+5x^{2}+5x+4 $.

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.

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

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

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

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 } $.

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

Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 1 } $

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

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

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

Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 1 } $

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

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.

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

Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrr}-4&1&5&5&\color{orangered}{ 4 }\\& & -4& -4& \color{orangered}{-4} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{1}&\color{orangered}{0} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.

Synthetic Division Calculator