Divide using synthetic division $$ ( x^{4}-4x^{3}-3x^{2}+16x-4 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&1&-4&-3&16&-4\\& & -2& 12& -18& \color{black}{4} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{9}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-4x^{3}-3x^{2}+16x-4 }{ x+2 } = \color{blue}{x^{3}-6x^{2}+9x-2} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-4&-3&16&-4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 1 }&-4&-3&16&-4\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-4&-3&16&-4\\& & \color{blue}{-2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-2&1&\color{orangered}{ -4 }&-3&16&-4\\& & \color{orangered}{-2} & & & \\ \hline &1&\color{orangered}{-6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-4&-3&16&-4\\& & -2& \color{blue}{12} & & \\ \hline &1&\color{blue}{-6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 12 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-2&1&-4&\color{orangered}{ -3 }&16&-4\\& & -2& \color{orangered}{12} & & \\ \hline &1&-6&\color{orangered}{9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-4&-3&16&-4\\& & -2& 12& \color{blue}{-18} & \\ \hline &1&-6&\color{blue}{9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-2&1&-4&-3&\color{orangered}{ 16 }&-4\\& & -2& 12& \color{orangered}{-18} & \\ \hline &1&-6&9&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-4&-3&16&-4\\& & -2& 12& -18& \color{blue}{4} \\ \hline &1&-6&9&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&1&-4&-3&16&\color{orangered}{ -4 }\\& & -2& 12& -18& \color{orangered}{4} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{9}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-6x^{2}+9x-2 } $ with a remainder of $ \color{red}{ 0 } $.