Divide using synthetic division $$ ( 35x^{2}-36x-34 ) \div ( 7x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&35&-36&-34\\& & -140& \color{black}{704} \\ \hline &\color{blue}{35}&\color{blue}{-176}&\color{orangered}{670} \end{array} $$The solution is:
$$ \frac{ 35x^{2}-36x-34 }{ x+4 } = \color{blue}{35x-176} ~+~ \frac{ \color{red}{ 670 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-4}&35&-36&-34\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 35 }&-36&-34\\& & & \\ \hline &\color{orangered}{35}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 35 } = \color{blue}{ -140 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&35&-36&-34\\& & \color{blue}{-140} & \\ \hline &\color{blue}{35}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ \left( -140 \right) } = \color{orangered}{ -176 } $
$$ \begin{array}{c|rrr}-4&35&\color{orangered}{ -36 }&-34\\& & \color{orangered}{-140} & \\ \hline &35&\color{orangered}{-176}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -176 \right) } = \color{blue}{ 704 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&35&-36&-34\\& & -140& \color{blue}{704} \\ \hline &35&\color{blue}{-176}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 704 } = \color{orangered}{ 670 } $
$$ \begin{array}{c|rrr}-4&35&-36&\color{orangered}{ -34 }\\& & -140& \color{orangered}{704} \\ \hline &\color{blue}{35}&\color{blue}{-176}&\color{orangered}{670} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 35x-176 } $ with a remainder of $ \color{red}{ 670 } $.