Divide using synthetic division $$ ( 3x^{2}+5x-17 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&3&5&-17\\& & -15& \color{black}{50} \\ \hline &\color{blue}{3}&\color{blue}{-10}&\color{orangered}{33} \end{array} $$The solution is:
$$ \frac{ 3x^{2}+5x-17 }{ x+5 } = \color{blue}{3x-10} ~+~ \frac{ \color{red}{ 33 } }{ x+5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&5&-17\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 3 }&5&-17\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&5&-17\\& & \color{blue}{-15} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrr}-5&3&\color{orangered}{ 5 }&-17\\& & \color{orangered}{-15} & \\ \hline &3&\color{orangered}{-10}& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&5&-17\\& & -15& \color{blue}{50} \\ \hline &3&\color{blue}{-10}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 50 } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrr}-5&3&5&\color{orangered}{ -17 }\\& & -15& \color{orangered}{50} \\ \hline &\color{blue}{3}&\color{blue}{-10}&\color{orangered}{33} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-10 } $ with a remainder of $ \color{red}{ 33 } $.