Check if $ x+2 $ is a factor of $ 4x^{6}-3x^{5}+2x^{4}+4x^{3}+2x^{2}+x+1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&4&-3&2&4&2&1&1\\& & -8& 22& -48& 88& -180& \color{black}{358} \\ \hline &\color{blue}{4}&\color{blue}{-11}&\color{blue}{24}&\color{blue}{-44}&\color{blue}{90}&\color{blue}{-179}&\color{orangered}{359} \end{array} $$Because the remainder $ \left( \color{red}{ 359 } \right) $ is not zero, we conclude that the $ x+2 $ is not a factor of $ 4x^{6}-3x^{5}+2x^{4}+4x^{3}+2x^{2}+x+1$.
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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&4&-3&2&4&2&1&1\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 4 }&-3&2&4&2&1&1\\& & & & & & & \\ \hline &\color{orangered}{4}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&4&-3&2&4&2&1&1\\& & \color{blue}{-8} & & & & & \\ \hline &\color{blue}{4}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrrrr}-2&4&\color{orangered}{ -3 }&2&4&2&1&1\\& & \color{orangered}{-8} & & & & & \\ \hline &4&\color{orangered}{-11}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&4&-3&2&4&2&1&1\\& & -8& \color{blue}{22} & & & & \\ \hline &4&\color{blue}{-11}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 22 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrrrr}-2&4&-3&\color{orangered}{ 2 }&4&2&1&1\\& & -8& \color{orangered}{22} & & & & \\ \hline &4&-11&\color{orangered}{24}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 24 } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&4&-3&2&4&2&1&1\\& & -8& 22& \color{blue}{-48} & & & \\ \hline &4&-11&\color{blue}{24}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -44 } $
$$ \begin{array}{c|rrrrrrr}-2&4&-3&2&\color{orangered}{ 4 }&2&1&1\\& & -8& 22& \color{orangered}{-48} & & & \\ \hline &4&-11&24&\color{orangered}{-44}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -44 \right) } = \color{blue}{ 88 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&4&-3&2&4&2&1&1\\& & -8& 22& -48& \color{blue}{88} & & \\ \hline &4&-11&24&\color{blue}{-44}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 88 } = \color{orangered}{ 90 } $
$$ \begin{array}{c|rrrrrrr}-2&4&-3&2&4&\color{orangered}{ 2 }&1&1\\& & -8& 22& -48& \color{orangered}{88} & & \\ \hline &4&-11&24&-44&\color{orangered}{90}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 90 } = \color{blue}{ -180 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&4&-3&2&4&2&1&1\\& & -8& 22& -48& 88& \color{blue}{-180} & \\ \hline &4&-11&24&-44&\color{blue}{90}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -180 \right) } = \color{orangered}{ -179 } $
$$ \begin{array}{c|rrrrrrr}-2&4&-3&2&4&2&\color{orangered}{ 1 }&1\\& & -8& 22& -48& 88& \color{orangered}{-180} & \\ \hline &4&-11&24&-44&90&\color{orangered}{-179}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -179 \right) } = \color{blue}{ 358 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&4&-3&2&4&2&1&1\\& & -8& 22& -48& 88& -180& \color{blue}{358} \\ \hline &4&-11&24&-44&90&\color{blue}{-179}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 358 } = \color{orangered}{ 359 } $
$$ \begin{array}{c|rrrrrrr}-2&4&-3&2&4&2&1&\color{orangered}{ 1 }\\& & -8& 22& -48& 88& -180& \color{orangered}{358} \\ \hline &\color{blue}{4}&\color{blue}{-11}&\color{blue}{24}&\color{blue}{-44}&\color{blue}{90}&\color{blue}{-179}&\color{orangered}{359} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 359 }\right)$.