Check if $ x+1 $ is a factor of $ 12x^{3}+37x^{2}-4x $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&12&37&-4&0\\& & -12& -25& \color{black}{29} \\ \hline &\color{blue}{12}&\color{blue}{25}&\color{blue}{-29}&\color{orangered}{29} \end{array} $$Because the remainder $ \left( \color{red}{ 29 } \right) $ is not zero, we conclude that the $ x+1 $ is not a factor of $ 12x^{3}+37x^{2}-4x$.
Explanation
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 + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&37&-4&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 12 }&37&-4&0\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 12 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&37&-4&0\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrr}-1&12&\color{orangered}{ 37 }&-4&0\\& & \color{orangered}{-12} & & \\ \hline &12&\color{orangered}{25}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 25 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&37&-4&0\\& & -12& \color{blue}{-25} & \\ \hline &12&\color{blue}{25}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -29 } $
$$ \begin{array}{c|rrrr}-1&12&37&\color{orangered}{ -4 }&0\\& & -12& \color{orangered}{-25} & \\ \hline &12&25&\color{orangered}{-29}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -29 \right) } = \color{blue}{ 29 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&12&37&-4&0\\& & -12& -25& \color{blue}{29} \\ \hline &12&25&\color{blue}{-29}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 29 } = \color{orangered}{ 29 } $
$$ \begin{array}{c|rrrr}-1&12&37&-4&\color{orangered}{ 0 }\\& & -12& -25& \color{orangered}{29} \\ \hline &\color{blue}{12}&\color{blue}{25}&\color{blue}{-29}&\color{orangered}{29} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 29 }\right)$.