Divide using synthetic division $$ ( 3x^{6}+2x^{4}-14x^{2}+x+1 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&3&0&2&0&-14&1&1\\& & -6& 12& -28& 56& -84& \color{black}{166} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{14}&\color{blue}{-28}&\color{blue}{42}&\color{blue}{-83}&\color{orangered}{167} \end{array} $$The solution is:
$$ \frac{ 3x^{6}+2x^{4}-14x^{2}+x+1 }{ x+2 } = \color{blue}{3x^{5}-6x^{4}+14x^{3}-28x^{2}+42x-83} ~+~ \frac{ \color{red}{ 167 } }{ x+2 } $$Explanation
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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&2&0&-14&1&1\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 3 }&0&2&0&-14&1&1\\& & & & & & & \\ \hline &\color{orangered}{3}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&2&0&-14&1&1\\& & \color{blue}{-6} & & & & & \\ \hline &\color{blue}{3}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrrr}-2&3&\color{orangered}{ 0 }&2&0&-14&1&1\\& & \color{orangered}{-6} & & & & & \\ \hline &3&\color{orangered}{-6}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&2&0&-14&1&1\\& & -6& \color{blue}{12} & & & & \\ \hline &3&\color{blue}{-6}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 12 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&\color{orangered}{ 2 }&0&-14&1&1\\& & -6& \color{orangered}{12} & & & & \\ \hline &3&-6&\color{orangered}{14}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 14 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&2&0&-14&1&1\\& & -6& 12& \color{blue}{-28} & & & \\ \hline &3&-6&\color{blue}{14}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -28 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&2&\color{orangered}{ 0 }&-14&1&1\\& & -6& 12& \color{orangered}{-28} & & & \\ \hline &3&-6&14&\color{orangered}{-28}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -28 \right) } = \color{blue}{ 56 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&2&0&-14&1&1\\& & -6& 12& -28& \color{blue}{56} & & \\ \hline &3&-6&14&\color{blue}{-28}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 56 } = \color{orangered}{ 42 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&2&0&\color{orangered}{ -14 }&1&1\\& & -6& 12& -28& \color{orangered}{56} & & \\ \hline &3&-6&14&-28&\color{orangered}{42}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 42 } = \color{blue}{ -84 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&2&0&-14&1&1\\& & -6& 12& -28& 56& \color{blue}{-84} & \\ \hline &3&-6&14&-28&\color{blue}{42}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -84 \right) } = \color{orangered}{ -83 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&2&0&-14&\color{orangered}{ 1 }&1\\& & -6& 12& -28& 56& \color{orangered}{-84} & \\ \hline &3&-6&14&-28&42&\color{orangered}{-83}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -83 \right) } = \color{blue}{ 166 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&2&0&-14&1&1\\& & -6& 12& -28& 56& -84& \color{blue}{166} \\ \hline &3&-6&14&-28&42&\color{blue}{-83}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 166 } = \color{orangered}{ 167 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&2&0&-14&1&\color{orangered}{ 1 }\\& & -6& 12& -28& 56& -84& \color{orangered}{166} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{14}&\color{blue}{-28}&\color{blue}{42}&\color{blue}{-83}&\color{orangered}{167} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{5}-6x^{4}+14x^{3}-28x^{2}+42x-83 } $ with a remainder of $ \color{red}{ 167 } $.