Divide using synthetic division $$ ( -x^{3}-x^{2}+21x+45 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&-1&-1&21&45\\& & -2& -6& \color{black}{30} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{blue}{15}&\color{orangered}{75} \end{array} $$The solution is:
$$ \frac{ -x^{3}-x^{2}+21x+45 }{ x-2 } = \color{blue}{-x^{2}-3x+15} ~+~ \frac{ \color{red}{ 75 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&-1&21&45\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ -1 }&-1&21&45\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&-1&21&45\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}2&-1&\color{orangered}{ -1 }&21&45\\& & \color{orangered}{-2} & & \\ \hline &-1&\color{orangered}{-3}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&-1&21&45\\& & -2& \color{blue}{-6} & \\ \hline &-1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrr}2&-1&-1&\color{orangered}{ 21 }&45\\& & -2& \color{orangered}{-6} & \\ \hline &-1&-3&\color{orangered}{15}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 15 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&-1&21&45\\& & -2& -6& \color{blue}{30} \\ \hline &-1&-3&\color{blue}{15}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 45 } + \color{orangered}{ 30 } = \color{orangered}{ 75 } $
$$ \begin{array}{c|rrrr}2&-1&-1&21&\color{orangered}{ 45 }\\& & -2& -6& \color{orangered}{30} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{blue}{15}&\color{orangered}{75} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}-3x+15 } $ with a remainder of $ \color{red}{ 75 } $.