Divide using synthetic division $$ ( -8x^{2}+16x-30 ) \div ( x-3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-8&16&-30\\& & -24& \color{black}{-24} \\ \hline &\color{blue}{-8}&\color{blue}{-8}&\color{orangered}{-54} \end{array} $$The solution is:
$$ \dfrac{ -8x^{2}+16x-30 }{ x-3 } = \color{blue}{-8x-8} \color{red}{~-~} \dfrac{ \color{red}{ 54 } }{ x-3 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{3}&-8&16&-30\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -8 }&16&-30\\& & & \\ \hline &\color{orangered}{-8}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-8&16&-30\\& & \color{blue}{-24} & \\ \hline &\color{blue}{-8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}3&-8&\color{orangered}{ 16 }&-30\\& & \color{orangered}{-24} & \\ \hline &-8&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-8&16&-30\\& & -24& \color{blue}{-24} \\ \hline &-8&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -54 } $
$$ \begin{array}{c|rrr}3&-8&16&\color{orangered}{ -30 }\\& & -24& \color{orangered}{-24} \\ \hline &\color{blue}{-8}&\color{blue}{-8}&\color{orangered}{-54} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -8x-8 } $ with a remainder of $ \color{red}{ -54 } $.