Divide using synthetic division $$ ( 9x^{2}+x-38 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&9&1&-38\\& & 18& \color{black}{38} \\ \hline &\color{blue}{9}&\color{blue}{19}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 9x^{2}+x-38 }{ x-2 } = \color{blue}{9x+19} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{2}&9&1&-38\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ 9 }&1&-38\\& & & \\ \hline &\color{orangered}{9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&9&1&-38\\& & \color{blue}{18} & \\ \hline &\color{blue}{9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 18 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrr}2&9&\color{orangered}{ 1 }&-38\\& & \color{orangered}{18} & \\ \hline &9&\color{orangered}{19}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 19 } = \color{blue}{ 38 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&9&1&-38\\& & 18& \color{blue}{38} \\ \hline &9&\color{blue}{19}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -38 } + \color{orangered}{ 38 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}2&9&1&\color{orangered}{ -38 }\\& & 18& \color{orangered}{38} \\ \hline &\color{blue}{9}&\color{blue}{19}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x+19 } $ with a remainder of $ \color{red}{ 0 } $.