Divide using synthetic division $$ ( 5x^{2}+12x+3 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}1&5&12&3\\& & 5& \color{black}{17} \\ \hline &\color{blue}{5}&\color{blue}{17}&\color{orangered}{20} \end{array} $$The solution is:
$$ \frac{ 5x^{2}+12x+3 }{ x-1 } = \color{blue}{5x+17} ~+~ \frac{ \color{red}{ 20 } }{ x-1 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{1}&5&12&3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}1&\color{orangered}{ 5 }&12&3\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 5 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&5&12&3\\& & \color{blue}{5} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 5 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrr}1&5&\color{orangered}{ 12 }&3\\& & \color{orangered}{5} & \\ \hline &5&\color{orangered}{17}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 17 } = \color{blue}{ 17 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&5&12&3\\& & 5& \color{blue}{17} \\ \hline &5&\color{blue}{17}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 17 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrr}1&5&12&\color{orangered}{ 3 }\\& & 5& \color{orangered}{17} \\ \hline &\color{blue}{5}&\color{blue}{17}&\color{orangered}{20} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x+17 } $ with a remainder of $ \color{red}{ 20 } $.