Divide using synthetic division $$ ( -x^{3}-x^{2}+5x-3 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&-1&-1&5&-3\\& & -3& -12& \color{black}{-21} \\ \hline &\color{blue}{-1}&\color{blue}{-4}&\color{blue}{-7}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ -x^{3}-x^{2}+5x-3 }{ x-3 } = \color{blue}{-x^{2}-4x-7} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&-1&5&-3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ -1 }&-1&5&-3\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&-1&5&-3\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}3&-1&\color{orangered}{ -1 }&5&-3\\& & \color{orangered}{-3} & & \\ \hline &-1&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&-1&5&-3\\& & -3& \color{blue}{-12} & \\ \hline &-1&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}3&-1&-1&\color{orangered}{ 5 }&-3\\& & -3& \color{orangered}{-12} & \\ \hline &-1&-4&\color{orangered}{-7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-1&-1&5&-3\\& & -3& -12& \color{blue}{-21} \\ \hline &-1&-4&\color{blue}{-7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}3&-1&-1&5&\color{orangered}{ -3 }\\& & -3& -12& \color{orangered}{-21} \\ \hline &\color{blue}{-1}&\color{blue}{-4}&\color{blue}{-7}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}-4x-7 } $ with a remainder of $ \color{red}{ -24 } $.