Divide using synthetic division $$ ( -x^{3}-x^{2}+21x+45 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&-1&-1&21&45\\& & -1& -2& \color{black}{19} \\ \hline &\color{blue}{-1}&\color{blue}{-2}&\color{blue}{19}&\color{orangered}{64} \end{array} $$The solution is:
$$ \frac{ -x^{3}-x^{2}+21x+45 }{ x-1 } = \color{blue}{-x^{2}-2x+19} ~+~ \frac{ \color{red}{ 64 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&-1&21&45\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ -1 }&-1&21&45\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&-1&21&45\\& & \color{blue}{-1} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}1&-1&\color{orangered}{ -1 }&21&45\\& & \color{orangered}{-1} & & \\ \hline &-1&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&-1&21&45\\& & -1& \color{blue}{-2} & \\ \hline &-1&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}1&-1&-1&\color{orangered}{ 21 }&45\\& & -1& \color{orangered}{-2} & \\ \hline &-1&-2&\color{orangered}{19}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 19 } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&-1&21&45\\& & -1& -2& \color{blue}{19} \\ \hline &-1&-2&\color{blue}{19}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 45 } + \color{orangered}{ 19 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrr}1&-1&-1&21&\color{orangered}{ 45 }\\& & -1& -2& \color{orangered}{19} \\ \hline &\color{blue}{-1}&\color{blue}{-2}&\color{blue}{19}&\color{orangered}{64} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}-2x+19 } $ with a remainder of $ \color{red}{ 64 } $.