Divide using synthetic division $$ ( -3x^{2}-9x+18 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&-3&-9&18\\& & -15& \color{black}{-120} \\ \hline &\color{blue}{-3}&\color{blue}{-24}&\color{orangered}{-102} \end{array} $$The solution is:
$$ \frac{ -3x^{2}-9x+18 }{ x-5 } = \color{blue}{-3x-24} \color{red}{~-~} \frac{ \color{red}{ 102 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{5}&-3&-9&18\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ -3 }&-9&18\\& & & \\ \hline &\color{orangered}{-3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-3&-9&18\\& & \color{blue}{-15} & \\ \hline &\color{blue}{-3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrr}5&-3&\color{orangered}{ -9 }&18\\& & \color{orangered}{-15} & \\ \hline &-3&\color{orangered}{-24}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ -120 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-3&-9&18\\& & -15& \color{blue}{-120} \\ \hline &-3&\color{blue}{-24}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -120 \right) } = \color{orangered}{ -102 } $
$$ \begin{array}{c|rrr}5&-3&-9&\color{orangered}{ 18 }\\& & -15& \color{orangered}{-120} \\ \hline &\color{blue}{-3}&\color{blue}{-24}&\color{orangered}{-102} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x-24 } $ with a remainder of $ \color{red}{ -102 } $.