Divide using synthetic division $$ ( -3x^{3}+11x^{2}+5x-4 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&-3&11&5&-4\\& & -12& -4& \color{black}{4} \\ \hline &\color{blue}{-3}&\color{blue}{-1}&\color{blue}{1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -3x^{3}+11x^{2}+5x-4 }{ x-4 } = \color{blue}{-3x^{2}-x+1} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{4}&-3&11&5&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ -3 }&11&5&-4\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&-3&11&5&-4\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}4&-3&\color{orangered}{ 11 }&5&-4\\& & \color{orangered}{-12} & & \\ \hline &-3&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&-3&11&5&-4\\& & -12& \color{blue}{-4} & \\ \hline &-3&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}4&-3&11&\color{orangered}{ 5 }&-4\\& & -12& \color{orangered}{-4} & \\ \hline &-3&-1&\color{orangered}{1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&-3&11&5&-4\\& & -12& -4& \color{blue}{4} \\ \hline &-3&-1&\color{blue}{1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}4&-3&11&5&\color{orangered}{ -4 }\\& & -12& -4& \color{orangered}{4} \\ \hline &\color{blue}{-3}&\color{blue}{-1}&\color{blue}{1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{2}-x+1 } $ with a remainder of $ \color{red}{ 0 } $.