Divide using synthetic division $$ ( 4x^{2}-18x+4 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&4&-18&4\\& & 16& \color{black}{-8} \\ \hline &\color{blue}{4}&\color{blue}{-2}&\color{orangered}{-4} \end{array} $$The solution is:
$$ \frac{ 4x^{2}-18x+4 }{ x-4 } = \color{blue}{4x-2} \color{red}{~-~} \frac{ \color{red}{ 4 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&4&-18&4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 4 }&-18&4\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&4&-18&4\\& & \color{blue}{16} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 16 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrr}4&4&\color{orangered}{ -18 }&4\\& & \color{orangered}{16} & \\ \hline &4&\color{orangered}{-2}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&4&-18&4\\& & 16& \color{blue}{-8} \\ \hline &4&\color{blue}{-2}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrr}4&4&-18&\color{orangered}{ 4 }\\& & 16& \color{orangered}{-8} \\ \hline &\color{blue}{4}&\color{blue}{-2}&\color{orangered}{-4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x-2 } $ with a remainder of $ \color{red}{ -4 } $.