Divide using synthetic division $$ ( 3x^{6}+5x^{5}+9x^{2}+17x-1 ) \div ( 3x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-5&3&5&0&0&9&17&-1\\& & -15& 50& -250& 1250& -6295& \color{black}{31390} \\ \hline &\color{blue}{3}&\color{blue}{-10}&\color{blue}{50}&\color{blue}{-250}&\color{blue}{1259}&\color{blue}{-6278}&\color{orangered}{31389} \end{array} $$The solution is:
$$ \frac{ 3x^{6}+5x^{5}+9x^{2}+17x-1 }{ x+5 } = \color{blue}{3x^{5}-10x^{4}+50x^{3}-250x^{2}+1259x-6278} ~+~ \frac{ \color{red}{ 31389 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&5&0&0&9&17&-1\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-5&\color{orangered}{ 3 }&5&0&0&9&17&-1\\& & & & & & & \\ \hline &\color{orangered}{3}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&5&0&0&9&17&-1\\& & \color{blue}{-15} & & & & & \\ \hline &\color{blue}{3}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrrr}-5&3&\color{orangered}{ 5 }&0&0&9&17&-1\\& & \color{orangered}{-15} & & & & & \\ \hline &3&\color{orangered}{-10}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 50 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&5&0&0&9&17&-1\\& & -15& \color{blue}{50} & & & & \\ \hline &3&\color{blue}{-10}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 50 } = \color{orangered}{ 50 } $
$$ \begin{array}{c|rrrrrrr}-5&3&5&\color{orangered}{ 0 }&0&9&17&-1\\& & -15& \color{orangered}{50} & & & & \\ \hline &3&-10&\color{orangered}{50}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 50 } = \color{blue}{ -250 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&5&0&0&9&17&-1\\& & -15& 50& \color{blue}{-250} & & & \\ \hline &3&-10&\color{blue}{50}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -250 \right) } = \color{orangered}{ -250 } $
$$ \begin{array}{c|rrrrrrr}-5&3&5&0&\color{orangered}{ 0 }&9&17&-1\\& & -15& 50& \color{orangered}{-250} & & & \\ \hline &3&-10&50&\color{orangered}{-250}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -250 \right) } = \color{blue}{ 1250 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&5&0&0&9&17&-1\\& & -15& 50& -250& \color{blue}{1250} & & \\ \hline &3&-10&50&\color{blue}{-250}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 1250 } = \color{orangered}{ 1259 } $
$$ \begin{array}{c|rrrrrrr}-5&3&5&0&0&\color{orangered}{ 9 }&17&-1\\& & -15& 50& -250& \color{orangered}{1250} & & \\ \hline &3&-10&50&-250&\color{orangered}{1259}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1259 } = \color{blue}{ -6295 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&5&0&0&9&17&-1\\& & -15& 50& -250& 1250& \color{blue}{-6295} & \\ \hline &3&-10&50&-250&\color{blue}{1259}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -6295 \right) } = \color{orangered}{ -6278 } $
$$ \begin{array}{c|rrrrrrr}-5&3&5&0&0&9&\color{orangered}{ 17 }&-1\\& & -15& 50& -250& 1250& \color{orangered}{-6295} & \\ \hline &3&-10&50&-250&1259&\color{orangered}{-6278}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -6278 \right) } = \color{blue}{ 31390 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&5&0&0&9&17&-1\\& & -15& 50& -250& 1250& -6295& \color{blue}{31390} \\ \hline &3&-10&50&-250&1259&\color{blue}{-6278}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 31390 } = \color{orangered}{ 31389 } $
$$ \begin{array}{c|rrrrrrr}-5&3&5&0&0&9&17&\color{orangered}{ -1 }\\& & -15& 50& -250& 1250& -6295& \color{orangered}{31390} \\ \hline &\color{blue}{3}&\color{blue}{-10}&\color{blue}{50}&\color{blue}{-250}&\color{blue}{1259}&\color{blue}{-6278}&\color{orangered}{31389} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{5}-10x^{4}+50x^{3}-250x^{2}+1259x-6278 } $ with a remainder of $ \color{red}{ 31389 } $.