Divide using synthetic division $$ ( -4x^{4}+2x^{3}-5x^{2}-1 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&-4&2&-5&0&-1\\& & -12& -30& -105& \color{black}{-315} \\ \hline &\color{blue}{-4}&\color{blue}{-10}&\color{blue}{-35}&\color{blue}{-105}&\color{orangered}{-316} \end{array} $$The solution is:
$$ \frac{ -4x^{4}+2x^{3}-5x^{2}-1 }{ x-3 } = \color{blue}{-4x^{3}-10x^{2}-35x-105} \color{red}{~-~} \frac{ \color{red}{ 316 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-4&2&-5&0&-1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ -4 }&2&-5&0&-1\\& & & & & \\ \hline &\color{orangered}{-4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-4&2&-5&0&-1\\& & \color{blue}{-12} & & & \\ \hline &\color{blue}{-4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}3&-4&\color{orangered}{ 2 }&-5&0&-1\\& & \color{orangered}{-12} & & & \\ \hline &-4&\color{orangered}{-10}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-4&2&-5&0&-1\\& & -12& \color{blue}{-30} & & \\ \hline &-4&\color{blue}{-10}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -35 } $
$$ \begin{array}{c|rrrrr}3&-4&2&\color{orangered}{ -5 }&0&-1\\& & -12& \color{orangered}{-30} & & \\ \hline &-4&-10&\color{orangered}{-35}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -35 \right) } = \color{blue}{ -105 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-4&2&-5&0&-1\\& & -12& -30& \color{blue}{-105} & \\ \hline &-4&-10&\color{blue}{-35}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -105 \right) } = \color{orangered}{ -105 } $
$$ \begin{array}{c|rrrrr}3&-4&2&-5&\color{orangered}{ 0 }&-1\\& & -12& -30& \color{orangered}{-105} & \\ \hline &-4&-10&-35&\color{orangered}{-105}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -105 \right) } = \color{blue}{ -315 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-4&2&-5&0&-1\\& & -12& -30& -105& \color{blue}{-315} \\ \hline &-4&-10&-35&\color{blue}{-105}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -315 \right) } = \color{orangered}{ -316 } $
$$ \begin{array}{c|rrrrr}3&-4&2&-5&0&\color{orangered}{ -1 }\\& & -12& -30& -105& \color{orangered}{-315} \\ \hline &\color{blue}{-4}&\color{blue}{-10}&\color{blue}{-35}&\color{blue}{-105}&\color{orangered}{-316} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{3}-10x^{2}-35x-105 } $ with a remainder of $ \color{red}{ -316 } $.