Check if $ x+2 $ is a factor of $ 12x^{3}+29x^{2}+8x-4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&12&29&8&-4\\& & -24& -10& \color{black}{4} \\ \hline &\color{blue}{12}&\color{blue}{5}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Because the remainder equals zero, we conclude that the $ x+2 $ is a factor of the $ 12x^{3}+29x^{2}+8x-4 $.
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 + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&29&8&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 12 }&29&8&-4\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 12 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&29&8&-4\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-2&12&\color{orangered}{ 29 }&8&-4\\& & \color{orangered}{-24} & & \\ \hline &12&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&29&8&-4\\& & -24& \color{blue}{-10} & \\ \hline &12&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}-2&12&29&\color{orangered}{ 8 }&-4\\& & -24& \color{orangered}{-10} & \\ \hline &12&5&\color{orangered}{-2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&29&8&-4\\& & -24& -10& \color{blue}{4} \\ \hline &12&5&\color{blue}{-2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-2&12&29&8&\color{orangered}{ -4 }\\& & -24& -10& \color{orangered}{4} \\ \hline &\color{blue}{12}&\color{blue}{5}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.