Divide using synthetic division $$ ( -4x^{2}-11x+2 ) \div ( x-3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-4&-11&2\\& & -12& \color{black}{-69} \\ \hline &\color{blue}{-4}&\color{blue}{-23}&\color{orangered}{-67} \end{array} $$The solution is:
$$ \dfrac{ -4x^{2}-11x+2 }{ x-3 } = \color{blue}{-4x-23} \color{red}{~-~} \dfrac{ \color{red}{ 67 } }{ x-3 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{3}&-4&-11&2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -4 }&-11&2\\& & & \\ \hline &\color{orangered}{-4}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-4&-11&2\\& & \color{blue}{-12} & \\ \hline &\color{blue}{-4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrr}3&-4&\color{orangered}{ -11 }&2\\& & \color{orangered}{-12} & \\ \hline &-4&\color{orangered}{-23}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ -69 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-4&-11&2\\& & -12& \color{blue}{-69} \\ \hline &-4&\color{blue}{-23}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -69 \right) } = \color{orangered}{ -67 } $
$$ \begin{array}{c|rrr}3&-4&-11&\color{orangered}{ 2 }\\& & -12& \color{orangered}{-69} \\ \hline &\color{blue}{-4}&\color{blue}{-23}&\color{orangered}{-67} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x-23 } $ with a remainder of $ \color{red}{ -67 } $.