Divide using synthetic division $$ ( -9x^{2}+14x-8 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&-9&14&-8\\& & -36& \color{black}{-88} \\ \hline &\color{blue}{-9}&\color{blue}{-22}&\color{orangered}{-96} \end{array} $$The solution is:
$$ \frac{ -9x^{2}+14x-8 }{ x-4 } = \color{blue}{-9x-22} \color{red}{~-~} \frac{ \color{red}{ 96 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&-9&14&-8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ -9 }&14&-8\\& & & \\ \hline &\color{orangered}{-9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&-9&14&-8\\& & \color{blue}{-36} & \\ \hline &\color{blue}{-9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrr}4&-9&\color{orangered}{ 14 }&-8\\& & \color{orangered}{-36} & \\ \hline &-9&\color{orangered}{-22}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ -88 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&-9&14&-8\\& & -36& \color{blue}{-88} \\ \hline &-9&\color{blue}{-22}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -88 \right) } = \color{orangered}{ -96 } $
$$ \begin{array}{c|rrr}4&-9&14&\color{orangered}{ -8 }\\& & -36& \color{orangered}{-88} \\ \hline &\color{blue}{-9}&\color{blue}{-22}&\color{orangered}{-96} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -9x-22 } $ with a remainder of $ \color{red}{ -96 } $.