Divide using synthetic division $$ ( 8x^{2}+33x-35 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&8&33&-35\\& & -40& \color{black}{35} \\ \hline &\color{blue}{8}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 8x^{2}+33x-35 }{ x+5 } = \color{blue}{8x-7} $$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.
$$ \begin{array}{c|rrr}\color{blue}{-5}&8&33&-35\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 8 }&33&-35\\& & & \\ \hline &\color{orangered}{8}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 8 } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&8&33&-35\\& & \color{blue}{-40} & \\ \hline &\color{blue}{8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 33 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrr}-5&8&\color{orangered}{ 33 }&-35\\& & \color{orangered}{-40} & \\ \hline &8&\color{orangered}{-7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&8&33&-35\\& & -40& \color{blue}{35} \\ \hline &8&\color{blue}{-7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -35 } + \color{orangered}{ 35 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-5&8&33&\color{orangered}{ -35 }\\& & -40& \color{orangered}{35} \\ \hline &\color{blue}{8}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x-7 } $ with a remainder of $ \color{red}{ 0 } $.