Divide using synthetic division $$ ( -12x^{2}+15x-18 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}6&-12&15&-18\\& & -72& \color{black}{-342} \\ \hline &\color{blue}{-12}&\color{blue}{-57}&\color{orangered}{-360} \end{array} $$The solution is:
$$ \frac{ -12x^{2}+15x-18 }{ x-6 } = \color{blue}{-12x-57} \color{red}{~-~} \frac{ \color{red}{ 360 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{6}&-12&15&-18\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}6&\color{orangered}{ -12 }&15&-18\\& & & \\ \hline &\color{orangered}{-12}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-12&15&-18\\& & \color{blue}{-72} & \\ \hline &\color{blue}{-12}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -57 } $
$$ \begin{array}{c|rrr}6&-12&\color{orangered}{ 15 }&-18\\& & \color{orangered}{-72} & \\ \hline &-12&\color{orangered}{-57}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -57 \right) } = \color{blue}{ -342 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-12&15&-18\\& & -72& \color{blue}{-342} \\ \hline &-12&\color{blue}{-57}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ \left( -342 \right) } = \color{orangered}{ -360 } $
$$ \begin{array}{c|rrr}6&-12&15&\color{orangered}{ -18 }\\& & -72& \color{orangered}{-342} \\ \hline &\color{blue}{-12}&\color{blue}{-57}&\color{orangered}{-360} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -12x-57 } $ with a remainder of $ \color{red}{ -360 } $.