Divide using synthetic division $$ ( 4x^{2}-2x-1 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}6&4&-2&-1\\& & 24& \color{black}{132} \\ \hline &\color{blue}{4}&\color{blue}{22}&\color{orangered}{131} \end{array} $$The solution is:
$$ \frac{ 4x^{2}-2x-1 }{ x-6 } = \color{blue}{4x+22} ~+~ \frac{ \color{red}{ 131 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{6}&4&-2&-1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}6&\color{orangered}{ 4 }&-2&-1\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 4 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&4&-2&-1\\& & \color{blue}{24} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 24 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrr}6&4&\color{orangered}{ -2 }&-1\\& & \color{orangered}{24} & \\ \hline &4&\color{orangered}{22}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 22 } = \color{blue}{ 132 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&4&-2&-1\\& & 24& \color{blue}{132} \\ \hline &4&\color{blue}{22}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 132 } = \color{orangered}{ 131 } $
$$ \begin{array}{c|rrr}6&4&-2&\color{orangered}{ -1 }\\& & 24& \color{orangered}{132} \\ \hline &\color{blue}{4}&\color{blue}{22}&\color{orangered}{131} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x+22 } $ with a remainder of $ \color{red}{ 131 } $.