Divide using synthetic division $$ ( 4x^{2}-3x-19 ) \div ( 2x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}1&4&-3&-19\\& & 4& \color{black}{1} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{orangered}{-18} \end{array} $$The solution is:
$$ \frac{ 4x^{2}-3x-19 }{ x-1 } = \color{blue}{4x+1} \color{red}{~-~} \frac{ \color{red}{ 18 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{1}&4&-3&-19\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}1&\color{orangered}{ 4 }&-3&-19\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 4 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&4&-3&-19\\& & \color{blue}{4} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 4 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrr}1&4&\color{orangered}{ -3 }&-19\\& & \color{orangered}{4} & \\ \hline &4&\color{orangered}{1}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&4&-3&-19\\& & 4& \color{blue}{1} \\ \hline &4&\color{blue}{1}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 1 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrr}1&4&-3&\color{orangered}{ -19 }\\& & 4& \color{orangered}{1} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{orangered}{-18} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x+1 } $ with a remainder of $ \color{red}{ -18 } $.