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