Divide using synthetic division $$ ( 6x^{2}+17x-45 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-9&6&17&-45\\& & -54& \color{black}{333} \\ \hline &\color{blue}{6}&\color{blue}{-37}&\color{orangered}{288} \end{array} $$The solution is:
$$ \frac{ 6x^{2}+17x-45 }{ x+9 } = \color{blue}{6x-37} ~+~ \frac{ \color{red}{ 288 } }{ x+9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-9}&6&17&-45\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-9&\color{orangered}{ 6 }&17&-45\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 6 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrr}\color{blue}{-9}&6&17&-45\\& & \color{blue}{-54} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -37 } $
$$ \begin{array}{c|rrr}-9&6&\color{orangered}{ 17 }&-45\\& & \color{orangered}{-54} & \\ \hline &6&\color{orangered}{-37}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -37 \right) } = \color{blue}{ 333 } $.
$$ \begin{array}{c|rrr}\color{blue}{-9}&6&17&-45\\& & -54& \color{blue}{333} \\ \hline &6&\color{blue}{-37}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -45 } + \color{orangered}{ 333 } = \color{orangered}{ 288 } $
$$ \begin{array}{c|rrr}-9&6&17&\color{orangered}{ -45 }\\& & -54& \color{orangered}{333} \\ \hline &\color{blue}{6}&\color{blue}{-37}&\color{orangered}{288} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-37 } $ with a remainder of $ \color{red}{ 288 } $.