Divide using synthetic division $$ ( x^{3}+4x^{2}-19x+14 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&1&4&-19&14\\& & 2& 12& \color{black}{-14} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+4x^{2}-19x+14 }{ x-2 } = \color{blue}{x^{2}+6x-7} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&4&-19&14\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 1 }&4&-19&14\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&4&-19&14\\& & \color{blue}{2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 2 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}2&1&\color{orangered}{ 4 }&-19&14\\& & \color{orangered}{2} & & \\ \hline &1&\color{orangered}{6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&4&-19&14\\& & 2& \color{blue}{12} & \\ \hline &1&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 12 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}2&1&4&\color{orangered}{ -19 }&14\\& & 2& \color{orangered}{12} & \\ \hline &1&6&\color{orangered}{-7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&4&-19&14\\& & 2& 12& \color{blue}{-14} \\ \hline &1&6&\color{blue}{-7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}2&1&4&-19&\color{orangered}{ 14 }\\& & 2& 12& \color{orangered}{-14} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+6x-7 } $ with a remainder of $ \color{red}{ 0 } $.