Divide using synthetic division $$ ( -2x^{2}+12x-19 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&-2&12&-19\\& & -8& \color{black}{16} \\ \hline &\color{blue}{-2}&\color{blue}{4}&\color{orangered}{-3} \end{array} $$The solution is:
$$ \frac{ -2x^{2}+12x-19 }{ x-4 } = \color{blue}{-2x+4} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{4}&-2&12&-19\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ -2 }&12&-19\\& & & \\ \hline &\color{orangered}{-2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&-2&12&-19\\& & \color{blue}{-8} & \\ \hline &\color{blue}{-2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrr}4&-2&\color{orangered}{ 12 }&-19\\& & \color{orangered}{-8} & \\ \hline &-2&\color{orangered}{4}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&-2&12&-19\\& & -8& \color{blue}{16} \\ \hline &-2&\color{blue}{4}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 16 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrr}4&-2&12&\color{orangered}{ -19 }\\& & -8& \color{orangered}{16} \\ \hline &\color{blue}{-2}&\color{blue}{4}&\color{orangered}{-3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x+4 } $ with a remainder of $ \color{red}{ -3 } $.